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गणित MCQ

समाकलन · MP Board (MPBSE) कक्षा 12 साइन्स (Madhya Pradesh - हिन्दी माध्यम). 389 questions.

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389 questions · 377 auto-graded MCQ + 12 self-marked written.

MCQ 11 Mark
$\int \cos x^{\circ} d x=$
  • A
    $\sin x^{\circ}+c$
  • B
    $\frac{\pi}{180} \sin x^{\circ}+c$
  • ✓
    $\frac{180}{\pi} \sin x^{\circ}+c$
  • D
    $\frac{\pi x}{180} \sin x^{\circ}+c$
Answer
Correct option: C.
$\frac{180}{\pi} \sin x^{\circ}+c$
(C)
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MCQ 21 Mark
$\int_0^{\pi / 2} \log \tan x d x=$
  • A
    $\frac{\pi}{2} \log \frac{1}{2}$
  • B
    $\frac{\pi}{2} \log 2$
  • C
    1
  • ✓
    $0$
Answer
Correct option: D.
$0$
(D)
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MCQ 31 Mark
$\int \frac{e^{2 x}-1}{e^{2 x}+1} d x=$
  • A
    $\log \left(e^{2 x}+1\right)+c$.
  • ✓
    $\log \left(e^x+e^{-x}\right)+c$
  • C
    $\log \left(e^x-e^{-x}\right)+c$
  • D
    $e^{2 x}+c$
Answer
Correct option: B.
$\log \left(e^x+e^{-x}\right)+c$
(B)
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MCQ 61 Mark
$\int \cot ^2 x d x=$
  • A
    $\cot x-x+c$
  • ✓
    $-\cot x-x+c$
  • C
    $-2 \cot x \cdot \operatorname{cosec}^2 x+c$
  • D
    $-\operatorname{cosec}^2 x+c$
Answer
Correct option: B.
$-\cot x-x+c$
(B)
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MCQ 71 Mark
$\int e^{2 \log \sec x} d x=$
  • A
    $e^{2 \log \sec x}+c$
  • B
    $\tan x+c$
  • ✓
    $\sec x \cdot \tan x+c$
  • D
    $\log \sec x+c$
Answer
Correct option: C.
$\sec x \cdot \tan x+c$
(C)
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MCQ 81 Mark
$\int \frac{e^x}{x}\left(1+x \log _e x\right) d x=$
  • A
    $\frac{e^x}{x}+c$
  • ✓
    $e^x \log x+c$
  • C
    $\log x+c$
  • D
    $\frac{e^x}{x} \log x+c$
Answer
Correct option: B.
$e^x \log x+c$
(B)
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MCQ 101 Mark
$\int_\alpha^\beta \varphi(x) d x+\int_\beta^\alpha \varphi(x) d x=$
  • A
    1
  • B
    $2 \int_\alpha^\beta \varphi(x) d x$
  • C
    $-2 \int_\beta^\alpha \varphi(x) d x$
  • ✓
    $0$
Answer
Correct option: D.
$0$
(D)
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MCQ 111 Mark
यदि $f(-x)=-f(x)$ तब $\int_{-a}^a f(x) d x=$
  • A
    $2 \int_0^a f(x) d x$
  • ✓
    $0$
  • C
    1
  • D
    -1
Answer
Correct option: B.
$0$
(B)
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MCQ 121 Mark
$\int \frac{d x}{a^2+x^2}=$
  • A
    $\frac{1}{a} \tan ^{-1} \frac{a}{x}+C$
  • B
    $\tan ^{-1} \frac{x}{a}+C$
  • ✓
    $\frac{1}{a} \tan ^{-1} \frac{x}{a}+C$
  • D
    $\frac{1}{a} \tan ^{-1} x+C \quad$
Answer
Correct option: C.
$\frac{1}{a} \tan ^{-1} \frac{x}{a}+C$
(C)
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MCQ 131 Mark
$\int \frac{x e^x}{(x+1)^2} d x=$
  • ✓
    $\frac{e^x}{(x+1)^2}+C$
  • B
    $\frac{-e^x}{x+1}+C$
  • C
    $\frac{e^x}{x+1}+C$
  • D
    $\frac{-e^x}{(x+1)^2}+C$
Answer
Correct option: A.
$\frac{e^x}{(x+1)^2}+C$
(A)
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MCQ 141 Mark
$\int x^2 \cdot e^{x^3} d x=$
  • A
    $e^{x^3}+C$
  • ✓
    $\frac{1}{3} e^{x^3}+C$
  • C
    $e^{x^2}+C$
  • D
    $\frac{1}{3} e^{x^2}+C$
Answer
Correct option: B.
$\frac{1}{3} e^{x^3}+C$
(B)
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MCQ 151 Mark
$\int \sqrt{1+\cos 2 x} d x=$
  • ✓
    $\sqrt{2} \cos x+C$
  • B
    $\sqrt{2} \sin x+C$
  • C
    $-\cos x-\sin x+C$
  • D
    $\sqrt{2} \sin \frac{x}{2}+C$
Answer
Correct option: A.
$\sqrt{2} \cos x+C$
(A)
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MCQ 161 Mark
$\int \frac{d x}{x+\sqrt{x}} d x$
  • A
    $\log x+\log (1+\sqrt{x})+C$
  • ✓
    $2 \log (1+\sqrt{x})+C$
  • C
    $\log (1+\sqrt{x})+C$
  • D
    $\log \sqrt{x}+C$
Answer
Correct option: B.
$2 \log (1+\sqrt{x})+C$
(B)
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MCQ 181 Mark
$\int_0^{\pi / 2} \sin ^2 x d x=$
  • ✓
    $\frac{\pi}{4}$
  • B
    $-\frac{\pi}{4}$
  • C
    $\frac{\pi}{2}$
  • D
    $-\frac{\pi}{2}$
Answer
Correct option: A.
$\frac{\pi}{4}$
(A)
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MCQ 191 Mark
$\int_0^{\pi / 4} \tan ^2 \theta d \theta=$
  • ✓
    $1-\frac{\pi}{4}$
  • B
    $1+\frac{\pi}{4}$
  • C
    $-1-\frac{\pi}{4}$
  • D
    $\frac{-\pi}{4}$
Answer
Correct option: A.
$1-\frac{\pi}{4}$
(A)
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MCQ 201 Mark
$\int \sin x d x=$
  • ✓
    $-\cos x+\mathrm{K}$
  • B
    $\sin x+K$
  • C
    $-\sin x+K$
  • D
    $\tan x+K$
Answer
Correct option: A.
$-\cos x+\mathrm{K}$
(A)
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MCQ 211 Mark
$\int \frac{\sec ^2 x}{\operatorname{cosec}^2 x} d x=\ldots$
  • A
    $\cot x+x=c$
  • B
    $\cot x-x+c$
  • ✓
    $\tan x-x+c$
  • D
    $\tan x+c$
Answer
Correct option: C.
$\tan x-x+c$
(C)
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MCQ 221 Mark
$\int x^2 \sin x^3 d x=\ldots$
  • A
    $\frac{1}{3} \cos x^3+c$
  • B
    $-\frac{1}{3} \cos x^3+c$
  • ✓
    $\frac{-1}{3} \cos x^3+c$
  • D
    $\frac{1}{2} \sin ^2 x^3+c$
Answer
Correct option: C.
$\frac{-1}{3} \cos x^3+c$
(C)
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MCQ 231 Mark
$\int \frac{d x}{\sqrt{2 x-x^2}}$ का मान है :
  • ✓
    $\sin ^{-1}(x-1)+c$
  • B
    $\sin ^{-1}(1+x)+c$
  • C
    $\sin ^{-1}\left(1+x^2\right)+c$
  • D
    $-\sqrt{2 x-x^2}+c$
Answer
Correct option: A.
$\sin ^{-1}(x-1)+c$
(A)
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MCQ 241 Mark
$\int \frac{2 d x}{\sqrt{x}(x+9)}=$
  • A
    $\tan ^{-1} \sqrt{x}+c$
  • B
    $\tan ^{-1}\left(\frac{\sqrt{x}}{3}\right)+c$
  • C
    $\frac{2}{3} \tan ^{-1}(\sqrt{x})+c$
  • ✓
    $\frac{2}{3} \tan ^{-1}\left(\frac{\sqrt{x}}{3}\right)+c$
Answer
Correct option: D.
$\frac{2}{3} \tan ^{-1}\left(\frac{\sqrt{x}}{3}\right)+c$
(D)
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MCQ 251 Mark
$\int \frac{2 d x}{\sqrt{1-4 x^2}}=$
  • A
    $\tan ^{-1}(2 x)+c$
  • B
    $\cot ^{-1}(2 x)+c$
  • C
    $\cos ^{-1}(2 x)+c$
  • ✓
    $\sin ^{-1}(2 x)+c$
Answer
Correct option: D.
$\sin ^{-1}(2 x)+c$
(D)
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MCQ 261 Mark
$\int e^x\left(1+\tan x+\tan ^2 x\right) d x=$.
  • A
    $e^x \cos x+c$
  • B
    $e^x \sin x+c$
  • ✓
    $e^x \tan x+c$
  • D
    $e^x \sec x+c$
Answer
Correct option: C.
$e^x \tan x+c$
(C)
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MCQ 271 Mark
$\int\left(\frac{\cos 2 \theta-1}{\cos 2 \theta+1}\right) d \theta=$
  • A
    $\tan \theta-\theta+c$
  • B
    $\theta+\tan \theta+c$
  • ✓
    $\theta-\tan \theta+c$
  • D
    $-\theta-\cot \theta+c$
Answer
Correct option: C.
$\theta-\tan \theta+c$
(C)
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MCQ 281 Mark
$\int \log _{10} x d x=$
  • A
    $\log _e 10 x \log _e\left(\frac{x}{e}\right)+c$
  • ✓
    $\log _{10} e \cdot x \log _e\left(\frac{x}{e}\right)+c$
  • C
    $(x-1) \log _e x+c$
  • D
    $\frac{1}{x}+c$
Answer
Correct option: B.
$\log _{10} e \cdot x \log _e\left(\frac{x}{e}\right)+c$
(B)
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MCQ 291 Mark
$\int \cos \sqrt{x} d x$ :
  • A
    $\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}$
  • B
    $\sqrt{x} \sin \sqrt{x}-\cos \sqrt{x}$
  • C
    $\frac{1}{2}[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]$
  • ✓
    $2[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]$
Answer
Correct option: D.
$2[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]$
(D)
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MCQ 301 Mark
$\int \frac{d x}{\sin x+\cos x}=$
  • A
    $\log \tan \left(\frac{\pi}{8}+\frac{x}{2}\right)+c$
  • B
    $\log \tan \left(\frac{\pi}{8}-\frac{x}{2}\right)+c$
  • ✓
    $\frac{1}{\sqrt{2}} \log \tan \left(\frac{\pi}{8}+\frac{x}{2}\right)+c$
  • D
    कोई नहीं (None)
Answer
Correct option: C.
$\frac{1}{\sqrt{2}} \log \tan \left(\frac{\pi}{8}+\frac{x}{2}\right)+c$
(C)
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MCQ 311 Mark
$\int_0^1 \frac{x}{1+x} d x=\ldots$
  • ✓
    $1-\log 2$
  • B
    $1+\log 2$
  • C
    2
  • D
    $\log 2$
Answer
Correct option: A.
$1-\log 2$
(A)
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MCQ 321 Mark
$\text { (5. } \int_0^1 e^x d x=\ldots$
  • ✓
    $e-1$
  • B
    $\frac{e-1}{e}$
  • C
    $\frac{e^2-1}{e}$
  • D
    $\frac{3^2-1}{2}$
Answer
Correct option: A.
$e-1$
(A)
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MCQ 331 Mark
$\int_0^1 \sqrt{x(1-x)} d x$
  • ✓
    $\frac{\pi}{8}$
  • B
    $\frac{\pi}{6}$
  • C
    $\frac{\pi}{2}$
  • D
    $\frac{\pi}{2}$
Answer
Correct option: A.
$\frac{\pi}{8}$
(A)
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MCQ 341 Mark
$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}=$
  • A
    $0$
  • B
    1
  • C
    $\frac{\pi}{2}$
  • ✓
    $\frac{\pi}{4}$
Answer
Correct option: D.
$\frac{\pi}{4}$
(D)
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MCQ 351 Mark
$\int_0^1 \tan ^{-1}\left(1-x+x^2\right) d x=$
  • A
    $\log 2$
  • B
    $\log \frac{1}{2}$
  • ✓
    $\pi \log 2$
  • D
    $\frac{\pi}{2} \log \frac{1}{2}$
Answer
Correct option: C.
$\pi \log 2$
(C)
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MCQ 361 Mark
$\int_1^5 \frac{x^2}{x^2-4} d x=$ ?
  • A
    $2-\log _e \frac{15}{7}$
  • ✓
    $2+\log _e \frac{15}{7}$
  • C
    $2+\log _e 3+4 \log _e 5$
  • D
    $2-\tan ^{-1} \frac{15}{7}$
Answer
Correct option: B.
$2+\log _e \frac{15}{7}$
(B)
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MCQ 371 Mark
$\int_0^{\pi / 4}(\sqrt{\tan x}+\sqrt{\cot x}) d x=$
  • ✓
    $\frac{\pi}{\sqrt{2}}$
  • B
    $\frac{\pi}{2 \sqrt{2}}$
  • C
    $\frac{\pi}{4 \sqrt{2}}$
  • D
    कोई नहीं
Answer
Correct option: A.
$\frac{\pi}{\sqrt{2}}$
(A)
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MCQ 381 Mark
$\int_0^\pi|\cot x| d x$ बराबर है :
  • A
    $\frac{\pi}{2}$
  • B
    1
  • C
    -1
  • ✓
    $-\frac{\pi}{2}$
Answer
Correct option: D.
$-\frac{\pi}{2}$
(D)
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MCQ 391 Mark
$\int_{-0}^{\pi / 2} \frac{(\sin x+\cos x)^2}{\sqrt{1+\sin ^2 x}} d x$ का मान है :
  • A
    $0$
  • ✓
    1
  • C
    2
  • D
    3
Answer
Correct option: B.
1
(B)
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MCQ 401 Mark
$\int_{-\pi / 2}^{\pi / 2} \sin ^9 x d x=$
  • A
    -1
  • ✓
    $0$
  • C
    1
  • D
    कोई नहीं (None)
Answer
Correct option: B.
$0$
(B)
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MCQ 411 Mark
$\int_1^1 \frac{1+\log x}{x}$ का मान क्या है ?
  • ✓
    $\frac{3}{2}$
  • B
    $\frac{1}{2}$
  • C
    $e$
  • D
    $\frac{1}{e}$
Answer
Correct option: A.
$\frac{3}{2}$
(A)
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MCQ 421 Mark
$\int_{-1}^1 \sin ^3 x \cos ^2 x d x$ का मान क्या है ?
  • ✓
    $0$
  • B
    1
  • C
    $\frac{1}{2}$
  • D
    2
Answer
Correct option: A.
$0$
(A)
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MCQ 431 Mark
$\int_{\pi / 6}^{\pi / 3} \frac{d x}{\sin 2 x}$ का मान है :
  • A
    $\frac{1}{2} \log (-1)$
  • B
    $\log (-1)$
  • C
    $\log 3$
  • ✓
    $\log \sqrt{3}$
Answer
Correct option: D.
$\log \sqrt{3}$
(D)
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MCQ 441 Mark
$\int_a^b f(x) d x+\int_c^b f(x) d x$ का मान है :
  • ✓
    $\int_a^b f(x) d x$
  • B
    $\int_c^a f(x) d x$
  • C
    $\int_{a+b}^{b+c} f(x) d x$
  • D
    कोई नहीं (None)
Answer
Correct option: A.
$\int_a^b f(x) d x$
(A)
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MCQ 451 Mark
$\int_0^a f(x) d x$ का मान है :
  • A
    $\int_0^a f(x-) d x$
  • ✓
    $\int_0^a f(a-x) d x$
  • C
    $\int_0^a f(a+x) d x$
  • D
    कोई नहीं (None)
Answer
Correct option: B.
$\int_0^a f(a-x) d x$
(B)
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MCQ 461 Mark
$\int_0^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x$ का मान क्या है ?
  • A
    1
  • B
    $\frac{\pi}{2}$
  • ✓
    $0$
  • D
    $-\frac{\pi}{2}$
Answer
Correct option: C.
$0$
(C)
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MCQ 471 Mark
$\int e^x \sec x(1+\tan x) d x$ बराबर है:
  • A
    $e^x \cos x+c$
  • ✓
    $e^x \sec x+c$
  • C
    $e^x \sin x+c$
  • D
    $e^x \tan x+c$
Answer
Correct option: B.
$e^x \sec x+c$
(B)
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MCQ 481 Mark
$\int \frac{x d x}{(x-1)(x-2)}$ बराबर है :
  • A
    $\log \left|\frac{(x-1)^2}{x-2}\right|+c$
  • ✓
    $\log \left|\frac{(x-2)^2}{x-1}\right|+c$
  • C
    $\log I(x-1)(x-2) \mid+c$
  • D
    $\log \left|\left(\frac{x-1}{x-2}\right)^2\right|+c$
Answer
Correct option: B.
$\log \left|\frac{(x-2)^2}{x-1}\right|+c$
(B)
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MCQ 491 Mark
$\int \frac{\sqrt{\tan x}}{\sin x \cos x} d x$ बराबर है :
  • A
    $2 \sqrt{\cot x}+e$.
  • B
    $\frac{\sqrt{\tan x}}{2}+c$
  • ✓
    $2 \sqrt{\tan x}+c$
  • D
    कोई नहीं (None)
Answer
Correct option: C.
$2 \sqrt{\tan x}+c$
(C)
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MCQ 501 Mark
$\int \frac{(1+\log x)^2}{x} d x=$
  • ✓
    $\frac{1}{3}(1+\log x)^3+c$
  • B
    $\frac{1}{2}(1+\log x)^2+c$
  • C
    $\log (\log \sqrt{1+x})+c$
  • D
    कोई नहीं (None)
Answer
Correct option: A.
$\frac{1}{3}(1+\log x)^3+c$
(A)
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MCQ 511 Mark
$\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^2 d x$ बराबर है :
  • A
    $\frac{x^2}{2}+\log e^x-2 x+c$
  • ✓
    $\frac{x^2}{2}+\log x+2 x+c$
  • C
    $\frac{x^2}{2}+\log x-2 x+c$
  • D
    कोई नहीं (None)
Answer
Correct option: B.
$\frac{x^2}{2}+\log x+2 x+c$
(B)
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MCQ 521 Mark
$x>a \int \frac{d x}{x^2-a^2}+k$
  • ✓
    $\frac{1}{2 a} \log \frac{x-a}{x+a}+k$
  • B
    $\frac{1}{2 a} \log \frac{x+a}{x-a}+k$
  • C
    $\frac{1}{a} \log \left(x^2-a^2\right)+k$
  • D
    $\log \left(x+\sqrt{x^2-a^2}\right)+k$
Answer
Correct option: A.
$\frac{1}{2 a} \log \frac{x-a}{x+a}+k$
(A)
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MCQ 531 Mark
$\int \frac{\cos 2 x d x}{(\sin x+\cos x)^2}$
  • A
    $-\frac{1}{\sin x+\cos x}+C$
  • ✓
    $\log |\sin x+\cos x|+C$
  • C
    $\log |\sin x-\cos x|+C$
  • D
    $\frac{1}{(\sin x+\cos x)^2}$
Answer
Correct option: B.
$\log |\sin x+\cos x|+C$
(B)
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MCQ 541 Mark
$\int_a^b x^5 d x=$
  • A
    $b^3-a^3$
  • ✓
    $\frac{b^6-a^6}{6}$
  • C
    $\frac{a^6-b^6}{6}$
  • D
    $a^5-b^5$
Answer
Correct option: B.
$\frac{b^6-a^6}{6}$
(B)
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MCQ 551 Mark
$\int \frac{d x}{1+\cos x}=$
  • ✓
    $\tan \frac{x}{2}+k$
  • B
    $\frac{1}{2} \tan \frac{x}{2}+k$
  • C
    $2 \tan \frac{x}{2}+k$
  • D
    $\tan ^2 \frac{x}{2}+k$
Answer
Correct option: A.
$\tan \frac{x}{2}+k$
(A)
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MCQ 561 Mark
$\int \frac{d x}{\sqrt{x}}=$
  • A
    $\sqrt{x}+k$
  • ✓
    $2 \sqrt{x}+k$
  • C
    $x+k$
  • D
    $\frac{2}{3} x^{3 / 2}+k$
Answer
Correct option: B.
$2 \sqrt{x}+k$
(B)
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MCQ 591 Mark
$\int \frac{\left(\tan ^{-1} x\right)^2}{1+x^2} d x=$
  • A
    1
  • B
    $\frac{\pi^3}{64}$
  • C
    $\frac{\pi^2}{192}$
  • ✓
    इनमें से कोई नहीं
Answer
Correct option: D.
इनमें से कोई नहीं
(D)
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MCQ 601 Mark
$\int \frac{(1+\log x)^2}{x} d x=$
  • ✓
    $\frac{1}{3}(1+\log x)^3+c$
  • B
    $\frac{1}{2}(1+\log x)^2+c$
  • C
    $\log (\log \overline{1+x})+c$
  • D
    इनमें से कोई नहीं
Answer
Correct option: A.
$\frac{1}{3}(1+\log x)^3+c$
(A)
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MCQ 611 Mark
$\int_{-\pi / 2}^{+\pi / 2} \sin ^9 x d x$ बराबर होगा :
  • A
    -1
  • B
    1
  • ✓
    $0$
  • D
    इनमें से कोई नहीं
Answer
Correct option: C.
$0$
(C)
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MCQ 621 Mark
$\int \frac{d x}{x^2+a^2}=$
  • ✓
    $\frac{1}{a} \tan ^{-1} \frac{x}{a}+k$
  • B
    $\frac{1}{a} \tan ^{-1}(x+a)+k$
  • C
    $\sin ^{-1} \frac{x}{a}+k$
  • D
    $\cos ^{-1} \frac{x}{a}+k$
Answer
Correct option: A.
$\frac{1}{a} \tan ^{-1} \frac{x}{a}+k$
(A)
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MCQ 631 Mark
$\int \frac{-1}{1+x^2} d x=$
  • A
    $\tan ^{-1} x+1$
  • B
    $\sec ^{-1} x+k$
  • C
    $\operatorname{cosec}^{-1} x+k$
  • ✓
    $\cot ^{-1} x+k$
Answer
Correct option: D.
$\cot ^{-1} x+k$
(D)
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MCQ 641 Mark
$\int_b^a x^2 d x=$
  • A
    $\frac{b^3-a^3}{3}$
  • ✓
    $\frac{a^3-b^3}{3}$
  • C
    $\frac{a^2-b^2}{2}$
  • D
    $\frac{b^2-a^2}{2}$
Answer
Correct option: B.
$\frac{a^3-b^3}{3}$
(B)
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MCQ 651 Mark
$\int \frac{d x}{1-\sin x}=$
  • A
    $\tan x-\sec x+k$
  • ✓
    $\tan x+\sec x+k$
  • C
    $\tan ^2 x+\sec ^2 x+k$
  • D
    $2(\tan x-\sec x)+k$
Answer
Correct option: B.
$\tan x+\sec x+k$
(B)
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MCQ 661 Mark
0 का $x$ के सापेक्ष समाकलन होता है :
  • A
    $0$
  • ✓
    $k$
  • C
    $x+k$
  • D
    $x^2+k$
Answer
Correct option: B.
$k$
(B)
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MCQ 671 Mark
$\int x^8 d x=$
  • A
    $8 x^7+k$
  • B
    $\frac{x^8}{8}+k$
  • C
    $x^9+k$
  • ✓
    $\frac{x^9}{9}+k$
Answer
Correct option: D.
$\frac{x^9}{9}+k$
(D)
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MCQ 681 Mark
$\int_a^b e^x d x=$
  • ✓
    $\left(e^b-e^a\right)$
  • B
    $e^{a-b}$
  • C
    $e^{-(a-b)}$
  • D
    $e^{a+b}$
Answer
Correct option: A.
$\left(e^b-e^a\right)$
(A)
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MCQ 691 Mark
$\int \frac{d x}{1+x^2}=$
  • A
    $\cot ^{-1} x+C$
  • ✓
    $\tan ^{-1} x+C$
  • C
    $\sec ^{-1} x+C$
  • D
    $\operatorname{cosec}^{-1}+C$
Answer
Correct option: B.
$\tan ^{-1} x+C$
(B)
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MCQ 701 Mark
$\int\left(\sin ^{-1} x+\cos ^{-1} x\right) d x=$
  • A
    $\frac{\pi}{2}+C$
  • ✓
    $\frac{\pi}{2} x+C$
  • C
    $\frac{2}{\sqrt{1-x^2}}+C$
  • D
    $x+C$
Answer
Correct option: B.
$\frac{\pi}{2} x+C$
(B)
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MCQ 711 Mark
$\int \sec x d x=$
  • A
    $\log \sec x$
  • ✓
    $\log [\sec x+\tan x]$
  • C
    $\log (\sec x-\tan x)$
  • D
    $\sec x \cdot \tan x$
Answer
Correct option: B.
$\log [\sec x+\tan x]$
(B)
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MCQ 721 Mark
$\int x^n d x, n \neq 0=$
  • A
    $\frac{x^{n-1}}{n-1}+C$
  • ✓
    $\frac{x^{n+1}}{n+1}+C$
  • C
    $x^{n+1}+C$
  • D
    $x^{n-1}+C$
Answer
Correct option: B.
$\frac{x^{n+1}}{n+1}+C$
(B)
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MCQ 731 Mark
$\int_a^b x^3 d x=\ldots \ldots$
  • A
    $\frac{b^3-a^3}{3}$
  • ✓
    $\frac{b^4-a^4}{4}$
  • C
    $\frac{b^2-a^2}{2}$
  • D
    $0$
Answer
Correct option: B.
$\frac{b^4-a^4}{4}$
(B)
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MCQ 741 Mark
$\int \frac{d x}{x}=\ldots \ldots$
  • A
    $x+k$
  • B
    $\frac{1}{x^2}+k$
  • C
    $-\frac{1}{x^2}+k$
  • ✓
    $\log x+k$
Answer
Correct option: D.
$\log x+k$
(D)
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MCQ 761 Mark
$\int x^5 d x$
  • ✓
    $\frac{x^6}{6}+k$
  • B
    $\frac{x^5}{5}+k$
  • C
    $\frac{x^7}{7}+k$
  • D
    $\frac{x^8}{8}+k$
Answer
Correct option: A.
$\frac{x^6}{6}+k$
(A)
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MCQ 771 Mark
यदि $\int \frac{d x}{(x+2)\left(x^2+1\right)}=a \log \left|1+x^2\right|+$$b \tan ^{-1} x+\frac{1}{5} \log |x+2|+C$ है, तो
  • A
    $a=\frac{-1}{10}, b=\frac{-2}{5}$
  • B
    $a=\frac{1}{10}, b=-\frac{2}{5}$
  • ✓
    $a=\frac{-1}{10}, b=\frac{2}{5}$
  • D
    $a=\frac{1}{10}, b=\frac{2}{5}$
Answer
Correct option: C.
$a=\frac{-1}{10}, b=\frac{2}{5}$
C
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MCQ 781 Mark
$\int \tan ^{-1} \sqrt{x} d x$ बराबर है
  • ✓
    $(x+1) \tan ^{-1} \sqrt{x}-\sqrt{x}+c$
  • B
    $x \tan ^{-1} \sqrt{x}-\sqrt{x}+c$
  • C
    $\sqrt{x}-x \tan ^{-1} \sqrt{x}+c$
  • D
    $\sqrt{x}-(x+1) \tan ^{-1} \sqrt{x}+c$
Answer
Correct option: A.
$(x+1) \tan ^{-1} \sqrt{x}-\sqrt{x}+c$
A
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MCQ 791 Mark
$\int \frac{x+\sin x}{1+\cos x} d x$ बराबर है
  • A
    $\log |1+\cos x|+c$
  • B
    $\log |x+\sin x|+c$
  • C
    $x-\tan \frac{x}{2}+c$
  • ✓
    $x \cdot \tan \frac{x}{2}+c$
Answer
Correct option: D.
$x \cdot \tan \frac{x}{2}+c$
D
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MCQ 801 Mark
$\int \frac{x^9}{\left(4 x^2+1\right)^6} d x$ बराबर है
  • A
    $\frac{1}{5 x}\left(4+\frac{1}{x^2}\right)^{-5}+C$
  • B
    $\frac{1}{5}\left(4+\frac{1}{x^2}\right)^{-5}+C$
  • C
    $\frac{1}{10 x}(1+4)^{-5}+C$
  • ✓
    $\frac{1}{10}\left(\frac{1}{x^2}+4\right)^{-5}+C$
Answer
Correct option: D.
$\frac{1}{10}\left(\frac{1}{x^2}+4\right)^{-5}+C$
D
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MCQ 811 Mark
$\int \frac{x^3}{x+1} d x$ बराबर है
  • A
    $x+\frac{x^2}{2}+\frac{x^3}{3}-\log |1-x|+C$
  • B
    $x+\frac{x^2}{2}-\frac{x^3}{3}-\log |1-x|+C$
  • C
    $x-\frac{x^2}{2}-\frac{x^3}{3}-\log |1+x|+C$
  • ✓
    $x-\frac{x^2}{2}+\frac{x^3}{3}-\log |1+x|+C$
Answer
Correct option: D.
$x-\frac{x^2}{2}+\frac{x^3}{3}-\log |1+x|+C$
D
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MCQ 821 Mark
$\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \frac{d x}{1+\cos 2 x}$ बराबर है
  • ✓
    1
  • B
    2
  • C
    3
  • D
    4
Answer
Correct option: A.
1
A
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MCQ 831 Mark
$\int \frac{d x}{\sin (x-a) \sin (x-b)}$ बराबर है
  • A
    $\sin (b-a)^{\log }\left|\frac{\sin (x-b)}{\sin (x-a)}\right|+C$
  • B
    $\operatorname{cosec}(b-a)^{\log }\left|\frac{\sin (x-a)}{\sin (x-b)}\right|+C$
  • ✓
    $\operatorname{cosec}(b-a) \log \left|\frac{\sin (x-b)}{\sin (x-a)}\right|+C$
  • D
    $\sin (b-a)^{\log }\left|\frac{\sin (x-a)}{\sin (x-b)}\right|+C$
Answer
Correct option: C.
$\operatorname{cosec}(b-a) \log \left|\frac{\sin (x-b)}{\sin (x-a)}\right|+C$
C
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MCQ 841 Mark
$\int \frac{\cos 2 x-\cos 2 \theta}{\cos x-\cos \theta} d x$ बराबर है
  • ✓
    $2(\sin x+x \cos \theta)+C$
  • B
    $2(\sin x-x \cos \theta)+C$
  • C
    $2(\sin x+2 x \cos \theta)+C$
  • D
    $2(\sin x-2 x \cos \theta)+C$
Answer
Correct option: A.
$2(\sin x+x \cos \theta)+C$
A
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MCQ 851 Mark
$\int e^x\left(\frac{1-x}{1+x^2}\right)^2 d x$ बराबर है
  • ✓
    $\frac{e^x}{1+x^2}+c$
  • B
    $\frac{-e^x}{1+x^2}+c$
  • C
    $\frac{e^x}{\left(1+x^2\right)^2}+c$
  • D
    $\frac{-e^x}{\left(1+x^2\right)^2}+c$
Answer
Correct option: A.
$\frac{e^x}{1+x^2}+c$
A
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MCQ 861 Mark
$\int_0^{\frac{\pi}{2}} \sqrt{1-\sin 2 x d x}$ बराबर है
  • A
    $2 \sqrt{2}$
  • B
    $2(\sqrt{2}+1)$
  • C
    2
  • ✓
    $2(\sqrt{2}-1)$
Answer
Correct option: D.
$2(\sqrt{2}-1)$
D
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MCQ 871 Mark
यदि $\int \frac{x^3 d x}{\sqrt{1+x^2}}=a\left(1+x^2\right)^{\frac{3}{2}}+b \sqrt{1+x^2}+c$, तब
  • A
    $a=\frac{1}{3}, b=1$
  • B
    $a=-\frac{1}{3}, b=1$
  • C
    $a=-\frac{1}{3}, b=-1$
  • ✓
    $a=\frac{1}{3}, b=-1$
Answer
Correct option: D.
$a=\frac{1}{3}, b=-1$
D
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MCQ 881 Mark
$\int \frac{d x}{e^x+e^{-x}}$ बराबर है
  • ✓
    $\tan ^{-1}\left(e^x\right)+C$
  • B
    $\tan ^{-1}\left(e^{-x}\right)+C$
  • C
    $\log \left(e^x-e^{-x}\right)+C$
  • D
    $\log \left(e^x+e^{-x}\right)+C$
Answer
Correct option: A.
$\tan ^{-1}\left(e^x\right)+C$
A
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MCQ 891 Mark
$\int \frac{d x}{\sin ^2 x \cos ^2 x}$ बराबर है :
  • A
    $\tan x+\cot x+C$
  • ✓
    $\tan x-\cot x+C$
  • C
    $\tan x \cot x+C$
  • D
    $\tan x-\cot 2 x+C$
Answer
Correct option: B.
$\tan x-\cot x+C$
B
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MCQ 901 Mark
$\int_1^{\sqrt{3}} \frac{d x}{1+x^2}$ बराबर है :
  • A
    $\frac{\pi}{3}$
  • B
    $\frac{2 \pi}{3}$
  • C
    $\frac{\pi}{6}$
  • ✓
    $\frac{\pi}{12}$
Answer
Correct option: D.
$\frac{\pi}{12}$
D
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MCQ 911 Mark
$\int x^2 e^{x^3} d x$ बराबर है
  • ✓
    $\frac{e^{x^3}}{3}+C$
  • B
    $\frac{e^{x^2}}{3}+C$
  • C
    $\frac{e^{x^3}}{2}+C$
  • D
    $\frac{1}{2} e^{x^2}+C$
Answer
Correct option: A.
$\frac{e^{x^3}}{3}+C$
A
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MCQ 921 Mark
$\int \frac{\cos 2 x}{(\sin x+\cos x)^2}=d x$
  • A
    $\log |\sin x-\cos x|+C$
  • ✓
    $\log |\sin x+\cos x|+C$
  • C
    $-\frac{1}{\sin x+\cos x}+C$
  • D
    $\frac{1}{(\sin x+\cos x)^2}+C$
Answer
Correct option: B.
$\log |\sin x+\cos x|+C$
B
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MCQ 931 Mark
समाकलन $\int_{\frac{1}{3}}^1 \frac{\left(x-x^3\right)^{\frac{1}{3}}}{x^4} d x$ का मान है :
  • ✓
    6
  • B
    $0$
  • C
    3
  • D
    4
Answer
Correct option: A.
6
A
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MCQ 941 Mark
यदि $f(x)=\int_0^x t \sin t d t$, तब $f^{\prime}(x)$ है :
  • A
    $\cos x+x \sin x$
  • ✓
    $x \sin x$
  • C
    $x \cos x$
  • D
    $\sin x+x \cos x$
Answer
Correct option: B.
$x \sin x$
B
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MCQ 951 Mark
यदि $\frac{d}{d x}\{f(x)\}=4 x^3-\frac{3}{x^4}$, जिसमें $f(2)=0$ है तो $f(x)$ हैं
  • ✓
    $x^4+\frac{1}{x^3}-\frac{129}{8}$
  • B
    $x^3+\frac{1}{x^4}+\frac{129}{8}$
  • C
    $x^4+\frac{1}{x^3}+\frac{129}{8}$
  • D
    $x^3+\frac{1}{x^4}-\frac{129}{8}$
Answer
Correct option: A.
$x^4+\frac{1}{x^3}-\frac{129}{8}$
A
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MCQ 961 Mark
यदि $f(a+b-x)=f(x)$, तो $\int_a^b x f(x) d x$ बराबर है :
  • A
    $\frac{a+b}{2} \int_a^b f(b-x) d x$
  • B
    $\frac{a+b}{2} \int_a^b(b+x) d x$
  • C
    $\frac{b-a}{2} \int_a^b f(x) d x$
  • ✓
    $\frac{a+b}{2} \int_a^b f(x) d x$
Answer
Correct option: D.
$\frac{a+b}{2} \int_a^b f(x) d x$
D
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MCQ 971 Mark
$\int \sqrt{x^2-8 x+7} d x$ बराबर है
  • A
    $\frac{1}{2}(x-4) \sqrt{x^2-8 x+7}$$+9 \log \left|x-4+\sqrt{x^2-8 x+7}\right|+C$
  • B
    $\frac{1}{2}(x+4) \sqrt{x^2-8 x+7}$$+9 \log \left|x+4+\sqrt{x^2-8 x+7}\right|+C$
  • C
    $\frac{1}{2}(x-4) \sqrt{x^2-8 x+7}$$-3 \sqrt{2} \log \left|x-4+\sqrt{x^2-8 x+7}\right|+C$
  • ✓
    $\frac{1}{2}(x-4) \sqrt{x^2-8 x+7}$$-\frac{9}{2} \log \left|x-4+\sqrt{x^2-8 x+7}\right|+C$
Answer
Correct option: D.
$\frac{1}{2}(x-4) \sqrt{x^2-8 x+7}$$-\frac{9}{2} \log \left|x-4+\sqrt{x^2-8 x+7}\right|+C$
D
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MCQ 981 Mark
$\int \sqrt{1+x^2} d x$ बराबर है:
  • ✓
    $\frac{x}{2} \sqrt{1+x^2}+\frac{1}{2} \log \left|\left(x+\sqrt{1+x^2}\right)\right|+C$
  • B
    $\frac{2}{3}\left(1+x^2\right)^{\frac{3}{2}}+C$
  • C
    $\frac{2}{3} x\left(1+x^2\right)^{\frac{3}{2}}+C$
  • D
    $\frac{x^2}{2} \sqrt{1+x^2}+\frac{1}{2} x^2 \log \left|x+\sqrt{1+x^2}\right|+C$
Answer
Correct option: A.
$\frac{x}{2} \sqrt{1+x^2}+\frac{1}{2} \log \left|\left(x+\sqrt{1+x^2}\right)\right|+C$
A
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MCQ 991 Mark
$\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)$ का प्रतिअवकलज है
  • A
    $\frac{1}{3} x^{\frac{1}{3}}+2 x^{\frac{1}{2}}+C$
  • B
    $\frac{2}{3} x^{\frac{2}{3}}+\frac{1}{2} x^2+C$
  • ✓
    $\frac{2}{3} x^{\frac{3}{2}}+2 x^{\frac{1}{2}}+C$
  • D
    $\frac{3}{2} x^{\frac{3}{2}}+\frac{1}{2} x^{\frac{1}{2}}+C$
Answer
Correct option: C.
$\frac{2}{3} x^{\frac{3}{2}}+2 x^{\frac{1}{2}}+C$
C
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MCQ 1001 Mark
$\int \frac{x d x}{(x-1)(x-2)}$ बराबर है
  • A
    $\log \left|\frac{(x-1)^2}{(x-2)}\right|+C$
  • ✓
    $\log \left|\frac{(x-2)^2}{x-1}\right|+C$
  • C
    $\log \left|\left(\frac{x-1}{x-2}\right)^2\right|+C$
  • D
    $\log |(x-1)(x-2)|+C$
Answer
Correct option: B.
$\log \left|\frac{(x-2)^2}{x-1}\right|+C$
B
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MCQ 1011 Mark
$\int \frac{\sin ^2 x-\cos ^2 x}{\sin ^2 x \cos ^2 x} d x$ बराबर हैं
  • ✓
    $\tan x+\cot x+C$
  • B
    $\tan x+\operatorname{cosec} x+C$
  • C
    $-\tan x+\cot x+C$
  • D
    $\tan x+\sec x+C$
Answer
Correct option: A.
$\tan x+\cot x+C$
A
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MCQ 1021 Mark
$\int \frac{e^x(1+x)}{\cos ^2\left(e^x x\right)} d x$ बराबर है
  • A
    $-\cot \left(e x^x\right)+C$
  • ✓
    $\tan \left(x e^x\right)+C$
  • C
    $\tan \left(e^x\right)+C$
  • D
    $\cot \left(e^x\right)+C$
Answer
Correct option: B.
$\tan \left(x e^x\right)+C$
B
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MCQ 1031 Mark
$\int \frac{d x}{x\left(x^2+1\right)}$ बराबर है
  • ✓
    $\log |x|-\frac{1}{2} \log \left(x^2+1\right)+C$
  • B
    $\log |x|+\frac{1}{2} \log \left(x^2+1\right)+C$
  • C
    $-\log |x|+\frac{1}{2} \log \left(x^2+1\right)+C$
  • D
    $\frac{1}{2} \log |x|+\log \left(x^2+1\right)+C$
Answer
Correct option: A.
$\log |x|-\frac{1}{2} \log \left(x^2+1\right)+C$
A
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MCQ 1041 Mark
$\int \frac{d x}{x^2+2 x+2}$ बराबर है:
  • A
    $x \tan ^{-1}(x+1)+C$
  • ✓
    $\tan ^{-1}(x+1)+C$
  • C
    $(x+1) \tan ^{-1} x+C$
  • D
    $\tan ^{-1} x+C$
Answer
Correct option: B.
$\tan ^{-1}(x+1)+C$
B
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MCQ 1051 Mark
$\int \frac{d x}{\sqrt{9 x-4 x^2}}$ बराबर है
  • A
    $\frac{1}{9} \sin ^{-1}\left(\frac{9 x-8}{8}\right)+C$
  • ✓
    $\frac{1}{2} \sin ^{-1}\left(\frac{8 x-9}{9}\right)+C$
  • C
    $\frac{1}{3} \sin ^{-1}\left(\frac{9 x-8}{8}\right)+C$
  • D
    $\frac{1}{2} \sin ^{-1}\left(\frac{9 x-8}{9}\right)+C$
Answer
Correct option: B.
$\frac{1}{2} \sin ^{-1}\left(\frac{8 x-9}{9}\right)+C$
B
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MCQ 1061 Mark
$\int \frac{10 x^9+10^x \log _e 10}{x^{10}+10^x} d x$ बराबर हैं
  • A
    $10^x-x^{10}+C$
  • B
    $10^x+x^{10}+C$
  • C
    $\left(10^x-x^{10}\right)^{-1}+C$
  • ✓
    $\log \left|10^x+x^{10}\right|+C$
Answer
Correct option: D.
$\log \left|10^x+x^{10}\right|+C$
D
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MCQ 1071 Mark
$\int e^x \sec x(1+\tan x) d x$ बराबर है
  • A
    $e^x \cos x+C$
  • ✓
    $e^x \sec x+C$
  • C
    $e^x \sin x+C$
  • D
    $e^x \tan x+C$
Answer
Correct option: B.
$e^x \sec x+C$
B
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MCQ 1081 Mark
$\int_0^{\frac{\pi}{2}} \log \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$ का मान है :
  • A
    2
  • B
    $\frac{3}{4}$
  • ✓
    $0$
  • D
    -2
Answer
Correct option: C.
$0$
C
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MCQ 1091 Mark
$\int_0^{\frac{2}{3}} \frac{d x}{4+9 x^2}$ बराबर है :
  • A
    $\frac{\pi}{6}$
  • B
    $\frac{\pi}{12}$
  • ✓
    $\frac{\pi}{24}$
  • D
    $\frac{\pi}{4}$
Answer
Correct option: C.
$\frac{\pi}{24}$
C
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MCQ 1101 Mark
$\int_0^1 \tan ^{-1}\left(\frac{2 x-1}{1+x-x^2}\right) d x$ का मान है :
  • A
    1
  • ✓
    $0$
  • C
    -1
  • D
    $\frac{\pi}{4}$
Answer
Correct option: B.
$0$
B
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MCQ 1111 Mark
$\int_{0}^{\frac{2}{3}} \frac{d x}{4+9 x^{2}}$ बराबर है:
  • A
    $\frac{\pi}{4}$
  • B
    $\frac{\pi}{6}$
  • C
    $\frac{\pi}{12}$
  • D
    $\frac{\pi}{24}$
Answer
$\int_{0}^{2 / 3} \frac{1}{4+9 x^{2}} d x$ $=\frac{1}{9} \int_{0}^{2 / 3} \frac{1}{\frac{4}{9}+x^{2}} d x$ $=\frac{1}{9} \int_{0}^{2 / 3} \frac{1}{\left(\frac{2}{3}\right)^{2}+x^{2}} d x$
$=\frac{1}{9} \cdot \frac{1}{\frac2 3}\left[\tan ^{-1}\left(\frac{x}{\frac2 3}\right)\right]_{0}^{\frac2 3}$ $\left(\because \int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \tan ^{-1} \frac{x}{a}\right)$
$=\frac{1}{6}\left[\tan ^{-1}\left(\frac{3 x}{2}\right)\right]_{0}^{\frac2 3}$ $=\frac{1}{6}\left[\tan ^{-1}\left(\frac{3}{2} \cdot \frac{2}{3}\right)-\tan ^{-1} 0\right]$ $=\frac{1}{6}\left(\tan ^{-1} 1-0\right)=\frac{1}{6} \cdot \frac{\pi}{4}=\frac{\pi}{24}$
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MCQ 1121 Mark
$\int_{1}^{\sqrt{3}} \frac{d x}{1+x^{2}}$ बराबर है:
  • A
    $\frac{\pi}{12}$
  • B
    $\frac{2\pi}{3}$
  • C
    $\frac{\pi}{6}$
  • D
    $\frac{\pi}{3}$
Answer
$\int_{1}^{\sqrt{3}} \frac{1}{1+x^{2}} d x$ $=\int_{1}^{\sqrt{3}} \frac{1}{x^{2}+1^{2}} d x$
$=\left[\frac{1}{1} \tan ^{-1}\left(\frac{x}{1}\right)\right]_{1}^{\sqrt{3}}$  = $\tan ^{-1} \sqrt{3}$  - $\tan ^{-1} 1\left(\because \int \frac{d x}{1+x^{2}}=\tan ^{-1} x\right)$
$=\frac{\pi}{3}-\frac{\pi}{4} \Rightarrow \frac{4 \pi-3 \pi}{12}=\frac{\pi}{12}$
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MCQ 1131 Mark
$\int \sqrt{x^{2}-8 x+7} d x$ बराबर है
  • A
    $\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}$ $+9 \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|$ + C
     
  • B
    $\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}$ $-\frac{9}{2} \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|$ + C
  • C
    $\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}$ $-3 \sqrt{2} \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|$ + C
  • D
    $\frac{1}{2}(x+4) \sqrt{x^{2}-8 x+7}$ $+9 \log \left|x+4+\sqrt{x^{2}-8 x+7}\right|+C$
Answer
माना $I=\int \sqrt{x^{2}-8 x+7} d x \Rightarrow I$ $=\int \sqrt{x^{2}-8 x+7+(4)^{2}-(4)^{2}} d x$
$=\int \sqrt{(x-4)^{2}+7-16} d x$ $=\int \sqrt{(x-4)^{2}-(3)^{2}} d x$ ($\because$ $\int \sqrt{x^{2}-a^{2}} d x$ $=\frac{x}{2} \sqrt{x^{2}-a^{2}}$ $-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|$
$\Rightarrow$ $I=\frac{x-4}{2} \sqrt{x^{2}-8 x+7}$ $-\frac{9}{2} \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|$ + C
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MCQ 1141 Mark
$\int \sqrt{1+x^{2}} d x$ बराबर है:
  • A
    $\frac{2}{3}\left(1+x^{2}\right)^{\frac{3}{2}}+C$
  • B
    $\frac{x^{2}}{2} \sqrt{1+x^{2}}$ $+\frac{1}{2} x^{2} \log \left|x+\sqrt{1+x^{2}}\right|$
  • C
    $\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log $ $\left|x+\sqrt{1+x^{2}}\right|+C$
  • D
    $\frac{2}{3} x\left(1+x^{2}\right)^{\frac{3}{2}}+C$
Answer
माना $I=\int \sqrt{1+x^{2}} d x$
$\Rightarrow$  $I=\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log $ $\left|x+\sqrt{1+x^{2}}\right|+C$ ($\because$ $\int \sqrt{x^{2}+a^{2}} d x$ $=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} $ $\log \left|x+\sqrt{x^{2}+a^{2}}\right|$)
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MCQ 1151 Mark
$\int \frac{d x}{x\left(x^{2}+1\right)}$ बराबर है।
  • A
    -log |x| + $\frac{1}{2}$log(x2 + 1) + C
  • B
    log |x| + $\frac{1}{2}$ log (x2 + 1) + C
  • C
    $\frac{1}{2}$ log |x| + log (x2 + 1) + C
  • D
    log |x| - $\frac{1}{2}$log (x2 + 1) + C
Answer
माना $\frac{1}{x\left(x^{2}+1\right)}$ $=\frac{A}{x}+\frac{B x+C}{x^{2}+1}$
$\Rightarrow$ 1 = A(x2 + 1) + (Bx + C)x
दोनों पक्षों में x2, x तथा अचर राशि के गुणांकों को समान रखने पर,
A + B = 0, C = 0 तथा A = 1
इन समीकरणों को सरल करने पर,
A = 1, B = -1 और C = 0
$\therefore$ $\frac{1}{x\left(x^{2}+1\right)}$ $=\frac{1}{x}+\frac{-x}{x^{2}+1}$
$\therefore$ $\int \frac{1}{x\left(x^{2}+1\right)} d x$ $=\int\left(\frac{1}{x}-\frac{x}{x^{2}+1} d x\right)$
= log |x| - $\frac{1}{2}$log (x2 + 1) + C (माना x2 =  1 = t $\Rightarrow$ 2x dx = dt $\Rightarrow$ x dx = $\frac{dt}{2}$, $\therefore$ $\int$$\frac{x}{x^2 + 1}$dx = $\int$$\frac{1}{t}$$\frac{dt}{2}$ = $\frac{1}{2}$log t)
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MCQ 1161 Mark
$\int \frac{x}{(x-1)(x-2)} d x$ बराबर है:
  • A
    log |(x - 1) (x - 2)| + C
  • B
    $\log \left|\frac{(x-2)^{2}}{x-1}\right|+C$
  • C
    $\log \left|\frac{(x-1)^{2}}{x-2}\right|+C$
  • D
    $\log \left|\left(\frac{x-1}{x-2}\right)^{2}\right|+\mathrm{C}$
Answer
माना $\frac{x}{(x-1)(x-2)}$ $=\frac{A}{(x-1)}+\frac{B}{(x-2)}$
$\Rightarrow$ x = A(x - 2) + B(x - 1) ...(i)
समी (i) में x = 1 और 2 रखने पर,
A = -1 और B = 2
$\therefore$ $\frac{x}{(x-1)(x-2)}$ $-\frac{1}{(x-1)}+\frac{2}{(x-2)}$
$\therefore$ $\int \frac{x}{(x-1)(x-2)} d x$ $=\int \frac{(-1)}{x-1} d x$ $+\int \frac{2}{x-2} d x$
= - log |x - 1| + 2 log |x - 2| + C
= - log |x - 1| + log |x - 2|2 + C
$=\log \left|\frac{(x-2)^{2}}{x-1}\right|+C$  ($\because$ log b - log a = log $\frac{b}{a}$)
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MCQ 1171 Mark
$\int \frac{1}{\sqrt{9 x-4 x^{2}}} d x$ के बराबर है:
  • A
    $\frac{1}{2} \sin ^{-1}\left(\frac{9 x-8}{9}\right)$ + C
  • B
    $\frac{1}{9} \sin ^{-1}\left(\frac{9 x-8}{8}\right)$ + C
  • C
    $\frac{1}{3} \sin ^{-1}\left(\frac{9 x-8}{8}\right)$ + C
  • D
    $\frac{1}{2} \sin ^{-1}\left(\frac{8 x-9}{9}\right)$ + C
Answer
$\int \frac{1}{\sqrt{9 x-4 x^{2}}} d x$ $=\frac{1}{\sqrt{4}} \int \frac{1}{\sqrt{\frac{9}{4} x-x^{2}}} d x$
$=\frac{1}{2} \int \frac{1}{\sqrt{-\left[x^{2}-\frac{9}{4} x+\left(\frac{9}{8}\right)^{2}\right]+\left(\frac{9}{8}\right)^{2}}} d x$ $=\frac{1}{2} \int \frac{1}{\sqrt{\left(\frac{9}{8}\right)^{2}-\left(x-\frac{9}{8}\right)^{2}}} d x$
$=\frac{1}{2} \sin ^{-1}\left(\frac{x-\frac{9}{8}}{\frac{9}{8}}\right)$ + C [$\because$ $\int \frac{d x}{\sqrt{a^{2}-x^{2}}}$ $=\sin ^{-1}\left(\frac{x}{a}\right)$]
$=\frac{1}{2} \sin ^{-1}\left(\frac{8 x-9}{9}\right)$ + C
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MCQ 1181 Mark
$\int \frac{1}{x^{2}+2 x+2} d x $ बराबर है:
  • A
    tan-1 x + C
  • B
    (x + 1) tan-1 x + C
  • C
    x tan-1 (x + 1) + C
  • D
    tan-1 (x + 1) + C
Answer
माना $I=\int \frac{1}{x^{2}+2 x+2} d x$ $=\int \frac{1}{(x+1)^{2}+1^{2}} d x$
माना x + 1 = t $\Rightarrow$ dx = dt
$\therefore$ $I=\int \frac{1}{t^{2}+1^{2}} d t$ $=\frac{1}{1} \tan ^{-1}\left(\frac{t}{1}\right)$ + C [$\because$ $\int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)$]
$=\tan ^{-1}\left(\frac{x+1}{1}\right)+C$ = tan−1 (x + 1) + C
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Question 1191 Mark
$\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)$ का प्रतिअवकलज है:
Answer
$\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x$ $=\int \sqrt{x} d x$ $+\int \frac{1}{\sqrt{x}} d x$ $=\int x^{\frac1 2} d x$ $+\int x^{-\frac1 2} d x$
$=\frac{x^{\frac3 2}}{\frac3 2}$ $+\frac{x^{\frac1 2}}{\frac1 2}+C$
$=\frac{2}{3} x^{\frac3 2}$ $+2 x^{\frac1 2}$ + C
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MCQ 1201 Mark
$\int ^{\frac{\pi}{2}}_{\frac{-\pi}{2}}$ (x3 + x cos x + tan5 x + 1)dx का मान है:
  • A
    $\pi$
  • B
    2
  • C
    1
Answer
माना $\int ^{ \frac{\pi}{2}}_{\frac{-\pi}{2}}$ (x3 + x cos x + tan5 x + 1)dx
$\Rightarrow$  $I=\int_{-\frac\pi 2}^{\frac\pi 2} x^{3} d x+\int_{-\frac\pi 2}^{\frac\pi 2} x \cos x d x$ $+\int_{-\frac\pi 2}^{\frac\pi 2} \tan ^{5} x d x+\int_{-\frac\pi 2}^{\frac\pi 2} 1 d x$
हम जानते हैं $\int_{-a}^{a}$ f(x)dx 
$\therefore$ I = 0 + 0 + 0 + 2 $\int_{0}^{\frac\pi 2}$ 1 dx [$\because$ x3, x cos x और tan5(x) विषम फलन हैं]
$\therefore$ $I=2[x]_{0}^{\frac\pi 2}=\frac{2 \pi}{2}=\pi{/tex}
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MCQ 1211 Mark
समाकलन $\int_{\frac{1}{3}}^{1} \frac{\left(x-x^{3}\right)^{\frac{1}{3}}}{x^{4}} d x$ का मान है:
  • A
    6
  • C
    3
  • D
    4
Answer
माना $I=\int_{\frac1 3}^{1} \frac{\left(x-x^{3}\right)^{\frac{1}{3}}}{x^{4}} d x$ $=\int_{\frac1 3}^{1} \frac{\left(x^{3}\right)^{\frac1 3}\left(\frac{x}{x^{3}}-\frac{x^{3}}{x^{3}}\right)^{\frac1 3}}{x^{4}} d x$ [अंश में $\left(x^{3}\right)^{\frac1 3}$ से गुणा और भाग करने पर]
$=\int_{\frac 1 3}^{1} \frac{\left(\frac{1}{x^{2}}-1\right)^{\frac1 3}}{x^{3}} d x \frac{1}{x^{2}}=t$ रखने पर,
$\Rightarrow$ $\frac{-2}{x^{3}} d x=d t \Rightarrow \frac{d x}{x^{3}}=\frac{d t}{(-2)}$
सीमाओं के लिए, जब x = 1 $\Rightarrow$ t = 1 और जब $x=\frac{1}{3} \Rightarrow t=3^{2}$ = 9 ($\because$ t = $\frac{1}{x^2}$)
$\therefore$ $I=\int_{9}^{1}(t-1)^{\frac1 3} \frac{d t}{(-2)}$ $=-\frac{1}{2}\left[\frac{(t-1)^{\frac{1}{3}}+1}{\frac{1}{3}+1}\right]_{9}^{1}$
$=-\frac{1}{2} \times \frac{3}{4}\left[(t-1)^{\frac{4}{3}}\right]_{9}^{1}$ $=-\frac{3}{8}\left[(1-1)^{\frac{4}{3}}-(9-1)^{\frac{4}{3}}\right]$
$=-\frac{3}{8}\left[0-\left(2^{3}\right)^{\frac{4}{3}}\right]$ $=-\frac{3}{8} \times(-16)=6$
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MCQ 1221 Mark
यदि f(x) = $\int_{0}^{x}$ t sin t dt, तब f'(x) है:
  • A
    x cos x
  • B
    x sin x
  • C
    cos x + x sin x
  • D
    sin x + x cos x
Answer
$f(x)=\int_{0}^{x} t \sin t d t$ 
$\Rightarrow \int \mathrm{u} . \mathrm{v} \mathrm{d} \mathrm{x}$ = $\mathrm{u} \cdot \int \mathrm{vdx}-\int \frac{\mathrm{du}}{\mathrm{dx}} \cdot\left\{\int \mathrm{vdx}\right\} \mathrm{d} \mathrm{x}$ 
f(x) = $[t]_{0}^{x} \int_{0}^{x} \sin t d t-\int_{0}^{x}\left\{\left(\frac{d}{d t} t\right) \cdot \int \sin t d t\right\} d t$ 
$=[t(-\cos t)]_{0}^{x}-\int_{0}^{x}(-\cos t) d t$ 
$=[-t(\cos t)+\sin t]_{0}^{x}$ 
= -x cos x + sin x - 0
$\Rightarrow$ f(x) = -x cos x + sin x
$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})$ = $-\left[\mathrm{x} \cdot \frac{\mathrm{d}}{\mathrm{dx}} \cos \mathrm{x}+\cos \mathrm{x} \cdot \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}+\frac{\mathrm{d}}{\mathrm{dx}} \sin \mathrm{x}\right]$ 
$\Rightarrow$ f'(x) = -[{x(-sin x)} + cos x] + cos x
= x sin x - cos x + cos x
= x sin x
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MCQ 1231 Mark
$\int x^m \cdot x^n d x=$
  • A
    $\frac{x^{m+1} \cdot x^{n+1}}{m+n+2}+k$
  • B
    $\frac{x^{m+n}}{m+n}+k$
  • ✓
    $\frac{x^{m+n+1}}{m+n+1}+k$
  • D
    $(m+n) x^{m+n-1}+k$
Answer
Correct option: C.
$\frac{x^{m+n+1}}{m+n+1}+k$
C
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MCQ 1241 Mark
$\int \tan \left(\tan ^{-1} x\right) d x=$
  • ✓
    $\frac{x^2}{2}+k$
  • B
    $\frac{x}{2}+k$
  • C
    $x+k$
  • D
    $\log \sec \left(\tan ^{-1} x\right)+k$
Answer
Correct option: A.
$\frac{x^2}{2}+k$
A
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MCQ 1251 Mark
$\int \log x^2 d x=$
  • A
    $\frac{1}{x^2}+k$
  • B
    $\frac{2}{x}+k$
  • C
    $x \log x-x+k$
  • ✓
    $2(x \log x-x)+k$
Answer
Correct option: D.
$2(x \log x-x)+k$
D
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MCQ 1261 Mark
$\int\left(\sin 3 x+4 \sin ^3 x\right) d x=$
  • A
    $3 \sin x+k$
  • ✓
    $-3 \cos x+k$
  • C
    $\frac{\cos 3 x}{3}+12 \sin ^2 x+k$
  • D
    $\frac{\cos 3 x}{3}+4 \cos ^3 x+k$
Answer
Correct option: B.
$-3 \cos x+k$
B
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MCQ 1271 Mark
$\int \frac{x-3}{x^2-9} d x=$
  • A
    $\log (x-3)+k$
  • ✓
    $\log (x+3)+k$
  • C
    $-\frac{1}{(x+3)^2}+k$
  • D
    $\frac{x^2}{2}-3 x+k$
Answer
Correct option: B.
$\log (x+3)+k$
B
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MCQ 1281 Mark
$\int \frac{(\sqrt{x}+1)^2}{x \sqrt{x}+2 x+\sqrt{x}} d x=$
  • A
    $\sqrt{x}+k$
  • B
    $\frac{1}{2} \sqrt{x}+k$
  • ✓
    $2 \sqrt{x}+k$
  • D
    $2 x+k$
Answer
Correct option: C.
$2 \sqrt{x}+k$
C
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MCQ 1291 Mark
$\int \frac{d x}{x \log x}=$
  • A
    $\log x+k$
  • B
    $(\log x)^2+k$
  • ✓
    $\log (\log x)+k$
  • D
    $\frac{1}{\log x}+k$
Answer
Correct option: C.
$\log (\log x)+k$
C
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MCQ 1301 Mark
$\int \frac{d x}{x^2+4}=$
  • A
    $\frac{1}{4} \tan ^{-1} \frac{x}{4}+k$
  • ✓
    $\frac{1}{2} \tan ^{-1} \frac{x}{2}+k$
  • C
    $\frac{1}{2} \tan ^{-1} \frac{2}{x}+k$
  • D
    $2 \tan ^{-1} \frac{x}{2}+k$
Answer
Correct option: B.
$\frac{1}{2} \tan ^{-1} \frac{x}{2}+k$
B
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MCQ 1311 Mark
$\int \frac{d x}{e^{-x}}=$
  • A
    $\frac{-1}{e^{-x}}+k$
  • ✓
    $e^x+k$
  • C
    $\frac{1}{e^{-x}} \cdot \frac{1}{x^2}+k$
  • D
    $-e^{-x}+k$
Answer
Correct option: B.
$e^x+k$
B
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MCQ 1321 Mark
$\int \frac{\cos 2 x}{\cos x+\sin x} d x=$
  • A
    $\sin x-\cos x+k$
  • B
    $-\sin x-\cos x+k$
  • ✓
    $\sin x+\cos x+k$
  • D
    $-\sin x+\cos x+k$
Answer
Correct option: C.
$\sin x+\cos x+k$
C
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MCQ 1331 Mark
$\int e^3 \cdot e^x d x=$
  • A
    $e^x+k$
  • B
    $\frac{e^{3+x}}{3}+k$
  • ✓
    $e^{x+3}+k$
  • D
    $3 e^{x+3}+k$
Answer
Correct option: C.
$e^{x+3}+k$
C
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MCQ 1341 Mark
$\int e^{2 \log x} d x=$
  • A
    $e^{2 \log x}+k$
  • B
    $\frac{x^2}{2}+k$
  • ✓
    $\frac{x^3}{3}+k$
  • D
    $3 x^3+k$
Answer
Correct option: C.
$\frac{x^3}{3}+k$
C
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MCQ 1351 Mark
$\int_\alpha^\beta \phi(x) d x+\int_\beta^\alpha \phi(x) d x=$
  • A
    2
  • B
    1
  • ✓
    $0$
  • D
    $2 \int_\alpha^\beta \phi(x) d x$
Answer
Correct option: C.
$0$
C
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MCQ 1361 Mark
$\int 2^{x+1} d x=$
  • ✓
    $\frac{2^{x+1}}{\log 2}+k$
  • B
    $2^{x+1} \log 2+k$
  • C
    $(x+1) 2^x+k$
  • D
    $2^{x+1}+k$
Answer
Correct option: A.
$\frac{2^{x+1}}{\log 2}+k$
A
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MCQ 1401 Mark
$\int_0^{\frac{\pi}{2}} \sin x \cdot \cos x d x$
  • A
    1
  • ✓
    $\frac{1}{2}$
  • C
    $-1$
  • D
    $\frac{1}{4}$
Answer
Correct option: B.
$\frac{1}{2}$
B
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MCQ 1411 Mark
$\int_0^{\frac{\pi}{2}} \log \tan x d x=$
  • A
    $\frac{\pi}{4}$
  • B
    $\frac{\pi}{2}$
  • ✓
    $0$
  • D
    $\pi$
Answer
Correct option: C.
$0$
C
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MCQ 1421 Mark
$\int_0^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=$
  • A
    $\pi$
  • B
    $\frac{\pi}{2}$
  • ✓
    $\frac{\pi}{4}$
  • D
    $2 \pi$
Answer
Correct option: C.
$\frac{\pi}{4}$
C
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MCQ 1431 Mark
$\int_0^a \frac{x d x}{2 \sqrt{a^2-x^2}}=$
  • A
    $\frac{a^2}{2}$
  • ✓
    $\frac{a}{2}$
  • C
    $\frac{a}{4}$
  • D
    $a$
Answer
Correct option: B.
$\frac{a}{2}$
B
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MCQ 1441 Mark
$\int_0^a \frac{d x}{\sqrt{x}}=$
  • A
    $2 \sqrt{x}$
  • ✓
    $2 \sqrt{a}$
  • C
    $\sqrt{x}$
  • D
    $\sqrt{a}$
Answer
Correct option: B.
$2 \sqrt{a}$
B
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MCQ 1461 Mark
$\int_0^1\left(x+2 x+3 x^2+4 x^3\right) d x=$
  • A
    10
  • B
    $\frac{5}{2}$
  • ✓
    $\frac{7}{2}$
  • D
    $\frac{1}{2}$
Answer
Correct option: C.
$\frac{7}{2}$
C
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MCQ 1471 Mark
$\int_0^1 \frac{4 \tan ^{-1} x}{1+x^2} d x=$
  • A
    $\frac{\pi^2}{4}$
  • ✓
    $\frac{\pi^2}{8}$
  • C
    $\frac{\pi}{4}$
  • D
    $\frac{\pi}{8}$
Answer
Correct option: B.
$\frac{\pi^2}{8}$
B
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MCQ 1521 Mark
$\int \sqrt{1+\cos 2 x} d x=$
  • A
    $\sqrt{2} \cos x+c$
  • ✓
    $\sqrt{2} \sin x+c$
  • C
    $\frac{2}{x^2}+c$
  • D
    $\sqrt{2} \sin \frac{x}{2}+c$
Answer
Correct option: B.
$\sqrt{2} \sin x+c$
B
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MCQ 1541 Mark
$\int(x+\cos 2 x) d x=$
  • A
    $\frac{1}{2} x \sin 2 x+\frac{1}{4} \cos 2 x+c$
  • B
    $\frac{1}{2} x \sin 2 x-\frac{1}{4} \cos 2 x+c$
  • C
    $2 x \sin 2 x+4 \cos 2 x+c$
  • ✓
    $\frac{x^2}{2}+\frac{\sin 2 x}{2}+c$
Answer
Correct option: D.
$\frac{x^2}{2}+\frac{\sin 2 x}{2}+c$
D
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MCQ 1551 Mark
$\int \tan ^2 x d x=$
  • A
    $\tan x+x+c$
  • ✓
    $\tan x-x+c$
  • C
    $\cot x+x+c$
  • D
    $\cot x-x+c$
Answer
Correct option: B.
$\tan x-x+c$
B
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MCQ 1561 Mark
$\int \sqrt{\cos x} \cdot \sin x d x=$
  • A
    $\frac{2}{3}(\cos x)^{\frac{3}{2}}+c$
  • ✓
    $-\frac{2}{3}(\cos x)^{\frac{3}{2}}+c$
  • C
    $(\cos x)^{3 / 2}+c$
  • D
    $-(\cos x)^{3 / 2}+c$
Answer
Correct option: B.
$-\frac{2}{3}(\cos x)^{\frac{3}{2}}+c$
B
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MCQ 1571 Mark
$\int \sqrt{a^2-x^2} d x=$
  • A
    $\frac{x}{2} \sqrt{a^2-x^2} d x$
  • B
    $\frac{a^2}{2} \sin ^{-1} \frac{x}{a}+c$
  • ✓
    $\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1} \frac{x}{a}+c$
  • D
    $\frac{x}{2} \sqrt{x^2-a^2}-\frac{a^2}{2} \sin ^{-1} \frac{x}{a}+c$
Answer
Correct option: C.
$\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1} \frac{x}{a}+c$
C
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MCQ 1581 Mark
$\int \sqrt{1-\sin 2 x} d x=$
  • ✓
    $\sin x+\cos x+c$
  • B
    $\sin x-\cos x+c$
  • C
    $\cos x-\sin x+c$
  • D
    $\tan x-\cot x+c$
Answer
Correct option: A.
$\sin x+\cos x+c$
A
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MCQ 1591 Mark
$\int \sec x d x=$
  • ✓
    $\log |\sec x+\tan x|+c$
  • B
    $\log |\sec x-\tan x|+c$
  • C
    $\log \sec x+c$
  • D
    $\tan ^5 x+c$
Answer
Correct option: A.
$\log |\sec x+\tan x|+c$
A
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MCQ 1601 Mark
$\int \sec ^5 x \tan x d x=$
  • A
    $5 \tan ^5 x+c$
  • ✓
    $\frac{1}{5} \sec ^5 x+c$
  • C
    $5 \log |\cos x|+c$
  • D
    $\tan ^5 x+c$
Answer
Correct option: B.
$\frac{1}{5} \sec ^5 x+c$
B
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MCQ 1631 Mark
$\int \frac{x^4+1}{x^2+1} d x=$
  • A
    $\frac{x^3}{3}+c$
  • ✓
    $\frac{x^3}{3}-x+2 \tan ^{-1} x+c$
  • C
    $2 \tan ^{-1} x+c$
  • D
    $\frac{x^3}{3}+x+2 \tan ^{-1} x+c$
Answer
Correct option: B.
$\frac{x^3}{3}-x+2 \tan ^{-1} x+c$
B
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MCQ 1641 Mark
$\int \frac{x^3 d x}{1+x^8}=$
  • A
    $\tan ^{-1} x^4+c$
  • B
    $4 \tan ^{-1} x^4+c$
  • ✓
    $\frac{1}{4} \tan ^{-1} x^4+c$
  • D
    $2 \tan ^{-1} x^4+c$
Answer
Correct option: C.
$\frac{1}{4} \tan ^{-1} x^4+c$
C
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MCQ 1651 Mark
$\int \frac{\log x}{x} d x=$
  • ✓
    $\frac{1}{2}(\log x)^2+c$
  • B
    $-\frac{1}{2}(\log x)^2+c$
  • C
    $\frac{2}{x^2}+c$
  • D
    $-\frac{2}{x^2}+c$
Answer
Correct option: A.
$\frac{1}{2}(\log x)^2+c$
A
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MCQ 1661 Mark
$\int \frac{d x}{x(x+2)}=$
  • A
    $\log \left|\frac{x}{x+2}\right|+c$
  • ✓
    $\frac{1}{2} \log \left|\frac{x}{x+2}\right|+c$
  • C
    $\log |x|+c$
  • D
    $\log |x+2|+c$
Answer
Correct option: B.
$\frac{1}{2} \log \left|\frac{x}{x+2}\right|+c$
B
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MCQ 1671 Mark
$\int \frac{d x}{a^2+x^2}=$
  • A
    $\frac{1}{a} \tan ^{-1} \frac{a}{x}+c$
  • B
    $\tan ^{-1} \frac{x}{a}+c$
  • ✓
    $\frac{1}{a} \tan ^{-1} \frac{x}{a}+c$
  • D
    $\frac{1}{a} \tan ^{-1} x+c$
Answer
Correct option: C.
$\frac{1}{a} \tan ^{-1} \frac{x}{a}+c$
C
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MCQ 1681 Mark
$\int \frac{d x}{2-3 x}=$
  • A
    $-3 \log |2-3 x|+c$
  • ✓
    $-\frac{1}{3} \log |2-3 x|+c$
  • C
    $-\log |2-3 x|+c$
  • D
    $2 \tan ^{-1} x^4+c$
Answer
Correct option: B.
$-\frac{1}{3} \log |2-3 x|+c$
B
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MCQ 1691 Mark
$\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x=$
  • ✓
    $2 \sin \sqrt{x}+c$
  • B
    $\sin \sqrt{x}+c$
  • C
    $\cos \sqrt{x}+c$
  • D
    $2 \cos \sqrt{x}+c$
Answer
Correct option: A.
$2 \sin \sqrt{x}+c$
A
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MCQ 1701 Mark
$\int \frac{\cos 2 x}{\cos ^2 x \cdot \sin ^2 x} d x=$
  • A
    $\cot x-\tan x+c$
  • B
    $\tan x-\cot x+c$
  • ✓
    $-\cot x-\tan x+c$
  • D
    $-\tan x+c$
Answer
Correct option: C.
$-\cot x-\tan x+c$
C
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MCQ 1711 Mark
$\int \frac{1-\cos 2 x}{1+\cos 2 x} d x=$
  • A
    $\tan x+c$
  • B
    $\tan x+x+c$
  • ✓
    $\tan x-x+c$
  • D
    $-\tan x+x+c$
Answer
Correct option: C.
$\tan x-x+c$
C
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MCQ 1721 Mark
$\int e^x\left\{\sin ^{-1} x+\frac{1}{\sqrt{1-x^2}}\right\} d x=$
  • A
    $e^x \cdot \frac{1}{\sqrt{1-x^2}}+c$
  • ✓
    $e^x \cdot \sin ^{-1} x+c$
  • C
    $\frac{e^x}{2}+c$
  • D
    $e^x \cdot \cos ^{-1} x+c$
Answer
Correct option: B.
$e^x \cdot \sin ^{-1} x+c$
B
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MCQ 1731 Mark
$\int \cot ^2 x d x=$
  • A
    $\cot x+x+k$
  • B
    $-\cot x+x+k$
  • ✓
    $-\cot x-x+k$
  • D
    $\cot x-x+k$
Answer
Correct option: C.
$-\cot x-x+k$
C
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MCQ 1741 Mark
$\int_4^9 \frac{1}{\sqrt{x}} d x=$
  • ✓
    2
  • B
    $\frac{\pi}{6}$
  • C
    $\frac{\pi}{4}$
  • D
    $\frac{\pi}{2}$
Answer
Correct option: A.
2
A
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MCQ 1761 Mark
$\int_0^{\frac{\pi}{6}} \cos x \cdot \cos 2 x d x=$
  • A
    $\frac{5}{6}$
  • B
    $\frac{1}{6}$
  • ✓
    $\frac{5}{12}$
  • D
    $-\frac{5}{12}$
Answer
Correct option: C.
$\frac{5}{12}$
C
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MCQ 1771 Mark
$\int_0^a \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} d x=$
  • A
    $a$
  • ✓
    $\frac{a}{2}$
  • C
    $2a$
  • D
    $3a$
Answer
Correct option: B.
$\frac{a}{2}$
B
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MCQ 1781 Mark
$\int x\left(4 x^2-6\right) d x=$
  • A
    $4 x^3-6 x+k$
  • B
    $\frac{4 x^4}{3}-6 x^2+k$
  • ✓
    $x^4-3 x^2+k$
  • D
    $\frac{4 x^3}{3}-3 x^2+k$
Answer
Correct option: C.
$x^4-3 x^2+k$
C
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MCQ 1791 Mark
$\int \sin \frac{3 x}{4} d x=$
  • A
    $k-\frac{3}{4} \cos \frac{3 x}{4}$
  • B
    $k+\frac{3}{4} \cos \frac{3 x}{4}$
  • ✓
    $k-\frac{4}{3} \cos \frac{3 x}{4}$
  • D
    $k+\frac{4}{3} \cos \frac{3 x}{4}$
Answer
Correct option: C.
$k-\frac{4}{3} \cos \frac{3 x}{4}$
C
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MCQ 1801 Mark
$\int \sin ^3 \theta \operatorname{cosec}^2 \theta d \theta=$
  • A
    $c+\theta$
  • B
    $c+\cos \theta$
  • ✓
    $c-\cos \theta$
  • D
    $c+\sin \theta$
Answer
Correct option: C.
$c-\cos \theta$
C
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MCQ 1811 Mark
$\int \sec ^2 \frac{17 x}{23} d x=$
  • A
    $k+\frac{17}{23} \tan \frac{17 x}{23}$
  • B
    $k-\frac{17}{23} \tan \frac{17 x}{23}$
  • ✓
    $k+\frac{23}{17} \tan \frac{17 x}{23}$
  • D
    $k-\frac{23}{17} \tan \frac{17 x}{23}$
Answer
Correct option: C.
$k+\frac{23}{17} \tan \frac{17 x}{23}$
C
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MCQ 1821 Mark
$\int \sec ^2 4 x d x=$
  • A
    $\tan 4 x+k$
  • ✓
    $\frac{1}{4} \tan 4 x+k$
  • C
    $4 \tan 4 x+k$
  • D
    $8 \tan 4 x+k$
Answer
Correct option: B.
$\frac{1}{4} \tan 4 x+k$
B
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MCQ 1831 Mark
$\int\left(\cos \theta \operatorname{cosec}^2 \theta-\cos \theta \cot ^2 \theta\right) d \theta=$
  • A
    $\log \operatorname{cosec} \theta+\cot \theta+k$
  • B
    $\operatorname{cosec} \theta \cot \theta+k$
  • ✓
    $k+\sin \theta$
  • D
    $\theta+k$
Answer
Correct option: C.
$k+\sin \theta$
C
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MCQ 1841 Mark
$\int \frac{x+2}{x^2-4} d x=$
  • A
    $\log |x+2|+k$
  • B
    $\log \left|x^2-4\right|+k$
  • ✓
    $\log |x-2|+k$
  • D
    $\log \left|\frac{x+2}{x-2}\right|+k$
Answer
Correct option: C.
$\log |x-2|+k$
C
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MCQ 1861 Mark
$\int \frac{d x}{x^2+5}$
  • A
    $\tan ^{-1} \frac{x}{5}+k$
  • B
    $\tan ^{-1} \frac{x}{\sqrt{5}}+k$
  • ✓
    $\frac{1}{\sqrt{5}} \tan ^{-1} \frac{x}{\sqrt{5}}+k$
  • D
    $\sqrt{5} \tan ^{-1} \frac{x}{\sqrt{5}}+k$
Answer
Correct option: C.
$\frac{1}{\sqrt{5}} \tan ^{-1} \frac{x}{\sqrt{5}}+k$
C
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MCQ 1871 Mark
$\int \frac{d x}{1+36 x^2}=$
  • A
    $6 \tan ^{-1} 6 x+k$
  • B
    $3 \tan ^{-1} 6 x+k$
  • ✓
    $\frac{1}{6} \tan ^{-1} 6 x+k$
  • D
    $\tan ^{-1} 6 x+k$
Answer
Correct option: C.
$\frac{1}{6} \tan ^{-1} 6 x+k$
C
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MCQ 1881 Mark
$\int \frac{3 x^2+2}{x^3+2 x} d x=$
  • A
    $\sin ^{-1}\left(x^3+2 x\right)+k$
  • B
    $\tan ^{-1}\left(3 x^2+2\right)+k$
  • C
    $\log \left|3 x^2+2\right|+k$
  • ✓
    $\log \left|x^3+2 x\right|+k$
Answer
Correct option: D.
$\log \left|x^3+2 x\right|+k$
D
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MCQ 1891 Mark
$\int \frac{3 d x}{\sqrt{1-9 x^2}}=$
  • A
    $\tan ^{-1} 3 x+k$
  • B
    $\sec ^{-1} 3 x+k$
  • ✓
    $\sin ^{-1} 3 x+k$
  • D
    $\cos ^{-1} 3 x+k$
Answer
Correct option: C.
$\sin ^{-1} 3 x+k$
C
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MCQ 1901 Mark
$\int \frac{3 \cos x-2 \sin x}{2 \cos x+3 \sin x} d x=$
  • A
    $2 \cos x+3 \sin x+k$
  • ✓
    $\log |2 \cos x+3 \sin x|+k$
  • C
    $\operatorname{tan}^{-1}\left(3 \sin \frac{x}{2}\right)+k$
  • D
    $2 \tan \frac{x}{2}+k$
Answer
Correct option: B.
$\log |2 \cos x+3 \sin x|+k$
B
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MCQ 1911 Mark
$\int e^x\left(x^3+3 x^2\right) d x=$
  • A
    $3 x^2 e^x+k$
  • B
    $x^2 e^x+k$
  • ✓
    $x^3 e^x+k$
  • D
    $3 e^x \cdot x^3+k$
Answer
Correct option: C.
$x^3 e^x+k$
C
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MCQ 1921 Mark
$\int e^x(\cos x-\sin x) d x=$
  • A
    $e^x \sin x+k$
  • ✓
    $e^x \cos x+k$
  • C
    $-e^x \sin x+k$
  • D
    $k-e^x \cos x$
Answer
Correct option: B.
$e^x \cos x+k$
B
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MCQ 1931 Mark
$\int \cos \frac{7 x}{9} d x=$
  • A
    $k+\sin \frac{7 x}{9}$
  • B
    $\frac{7}{9} \sin \frac{7 x}{9}+k$
  • ✓
    $\frac{9}{7} \sin \frac{7 x}{9}+k$
  • D
    $k+\cos \frac{7 x}{9}$
Answer
Correct option: C.
$\frac{9}{7} \sin \frac{7 x}{9}+k$
C
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MCQ 1941 Mark
$\int 4^x d x=$
  • A
    $4^x+k$
  • B
    $\frac{4^{x+1}}{x+1}+k$
  • ✓
    $\frac{4^x}{\log 4}+k$
  • D
    $-\frac{4}{\log 4}+k$
Answer
Correct option: C.
$\frac{4^x}{\log 4}+k$
C
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MCQ 1951 Mark
$\int(4 \cos x-5 \sin x) d x=$
  • ✓
    $k+4 \sin x+5 \cos x$
  • B
    $k-4 \sin x-5 \cos x$
  • C
    $k+4 \sin x-5 \cos x$
  • D
    $k-4 \sin x+5 \cos x$
Answer
Correct option: A.
$k+4 \sin x+5 \cos x$
A
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MCQ 1971 Mark
$\int_{-1}^1 \sin ^{17} x \cos ^3 x d x=$
  • A
    $\frac{12}{5}$
  • ✓
    $0$
  • C
    1
  • D
    $\frac{3}{5}$
Answer
Correct option: B.
$0$
B
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MCQ 1981 Mark
$\int_{-1}^1 \log \left(\frac{3+x}{3-x}\right) d x=$
  • ✓
    $0$
  • B
    1
  • C
    $2 \log 3$
  • D
    $3 \log 2$
Answer
Correct option: A.
$0$
A
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MCQ 1991 Mark
$3 \int_0^3 x^3 d x=$
  • A
    $\frac{81}{4}$
  • ✓
    $\frac{243}{4}$
  • C
    $0$
  • D
    $\frac{9}{4}$
Answer
Correct option: B.
$\frac{243}{4}$
B
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MCQ 2001 Mark
$25 \int \sec 5 x \tan 5 x d x=$
  • A
    $25 \sec 5 x+k$
  • ✓
    $5 \sec 5 x+k$
  • C
    $25 \tan 5 x+k$
  • D
    $\sec 5 x+k$
Answer
Correct option: B.
$5 \sec 5 x+k$
B
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MCQ 2011 Mark
$\int x e^x d x=?$
  • A
    $e^x(1-x)+C$
  • B
    $e^x(x+1)+C$
  • ✓
    $e^x(x-1)+C$
  • D
    इनमें से कोई नहीं
Answer
Correct option: C.
$e^x(x-1)+C$
C
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MCQ 2021 Mark
$\int \tan 10 x d x=$
  • A
    $k+\log |\cos 10 x|$
  • ✓
    $k+\frac{1}{10} \log |\sec 10 x|$
  • C
    $k+\log |\sec 10 x|$
  • D
    $\sec 10 x+k$
Answer
Correct option: B.
$k+\frac{1}{10} \log |\sec 10 x|$
B
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MCQ 2031 Mark
$\int \sin 6 x d x=$
  • A
    $k+\cos 6 x$
  • B
    $k+6 \cos 6 x$
  • C
    $k-6 \cos 6 x$
  • ✓
    $k-\frac{1}{6} \cos 6 x$
Answer
Correct option: D.
$k-\frac{1}{6} \cos 6 x$
D
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MCQ 2041 Mark
$\int\left[\frac{\sin ^2 x+\cos ^2 x}{\sec x}\right] d x=$
  • ✓
    $k+\sin x$
  • B
    $k-\sin x$
  • C
    $k+\sec x$
  • D
    $k-\sec x$
Answer
Correct option: A.
$k+\sin x$
A
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MCQ 2051 Mark
$\int\left(3 x^2+e^x\right) d x=$
  • ✓
    $x^3+e^x+k$
  • B
    $\frac{x^3}{3}+e^x+k$
  • C
    $\frac{x^3}{3}+\frac{e^x}{2}+k$
  • D
    $3 x^3+e^x +k$
Answer
Correct option: A.
$x^3+e^x+k$
A
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MCQ 2061 Mark
$\int \frac{x^2-4}{x-2} d x=$
  • A
    $\log \left|x^2-4\right|+k$
  • B
    $\log |x-2|+k$
  • C
    $\frac{x^2}{2}-x+k$
  • ✓
    $\frac{x^2}{2}+2 x+k$
Answer
Correct option: D.
$\frac{x^2}{2}+2 x+k$
D
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MCQ 2071 Mark
$\int_{\frac{\pi}{4}}^{\frac{\pi}{4}} \tan \theta d \theta=$
  • A
    $-\frac{1}{2} \log 2$
  • B
    $2 \log 2$
  • ✓
    $0$
  • D
    $5 \log 2$
Answer
Correct option: C.
$0$
C
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MCQ 2091 Mark
$\int \frac{d x}{x^2+81}=$
  • A
    $\tan ^{-1} \frac{x}{9}+k$
  • ✓
    $\frac{1}{9} \tan ^{-1} \frac{x}{9}+k$
  • C
    $9 \tan ^{-1} \frac{x}{9}+k$
  • D
    $\tan ^{-1}(x-9)+k$
Answer
Correct option: B.
$\frac{1}{9} \tan ^{-1} \frac{x}{9}+k$
B
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MCQ 2101 Mark
$\int \frac{d x}{7 x+5}=$
  • A
    $k+7 \log |7 x+5|$
  • ✓
    $k+\frac{1}{7} \log |7 x+5|$
  • C
    $k+\frac{1}{5} \log |7 x+5|$
  • D
    $k+\log |7 x+5|$
Answer
Correct option: B.
$k+\frac{1}{7} \log |7 x+5|$
B
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MCQ 2111 Mark
$\int \frac{6 x+5}{3 x^2+5 x+6} d x=$
  • ✓
    $\log \left|3 x^2+5 x+6\right|+k$
  • B
    $\log |6 x+5|+k$
  • C
    $\tan ^{-1}\left(3 x^2+5 x+6\right)+k$
  • D
    $\tan ^{-1}(6 x+5)+k$
Answer
Correct option: A.
$\log \left|3 x^2+5 x+6\right|+k$
A
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MCQ 2121 Mark
$\int \frac{5 d x}{\sqrt{1-25 x^2}}=$
  • A
    $\frac{1}{5} \sin ^{-1} 5 x+k$
  • B
    $\frac{1}{25} \sin ^{-1} 5 x+k$
  • ✓
    $\sin ^{-1} 5 x+k$
  • D
    $\sin ^{-1} \frac{x}{5}+k$
Answer
Correct option: C.
$\sin ^{-1} 5 x+k$
C
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MCQ 2131 Mark
$\int \frac{1}{x^3} d x=?$
  • A
    $\frac{-3}{x^2}+C$
  • ✓
    $\frac{-1}{2 x^2}+C$
  • C
    $\frac{-1}{3 x^2}+C$
  • D
    $\frac{x^{-2}}{2}+C$
Answer
Correct option: B.
$\frac{-1}{2 x^2}+C$
B
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MCQ 2161 Mark
$\int_{3 a}^{3 b} e^x d x=$
  • A
    $3\left(e^b-e^a\right)$
  • B
    $e^{3 b-3 a}$
  • ✓
    $e^{3 b}-e^{3 a}$
  • D
    $e^{3 b}+e^{3 a}$
Answer
Correct option: C.
$e^{3 b}-e^{3 a}$
C
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MCQ 2181 Mark
$\int_0^1 x(1-x)^9 d x=$
  • ✓
    $\frac{1}{110}$
  • B
    $\frac{1}{101}$
  • C
    $\frac{9}{101}$
  • D
    $\frac{113}{127}$
Answer
Correct option: A.
$\frac{1}{110}$
A
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MCQ 2191 Mark
$\int_0^{16} x^{\frac{3}{4}} d x=$
  • ✓
    $\frac{512}{7}$
  • B
    $\frac{256}{7}$
  • C
    $\frac{128}{7}$
  • D
    $\frac{64}{7}$
Answer
Correct option: A.
$\frac{512}{7}$
A
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MCQ 2201 Mark
$7 \int \frac{d x}{1+49 x^2}=$
  • A
    $\tan ^{-1}(49 x)+k$
  • ✓
    $\tan ^{-1}(7 x)+k$
  • C
    $7 \tan ^{-1}(49 x)+k$
  • D
    $\tan ^{-1} x+k$
Answer
Correct option: B.
$\tan ^{-1}(7 x)+k$
B
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MCQ 2221 Mark
$300 \int \sqrt{x} d x=$
  • A
    $\frac{2}{3} x^{\frac{3}{2}}+k$
  • B
    $300 x^{\frac{3}{2}}+k$
  • ✓
    $200 x^{\frac{3}{2}}+k$
  • D
    $\frac{200}{3} x^{\frac{3}{2}}+k$
Answer
Correct option: C.
$200 x^{\frac{3}{2}}+k$
C
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MCQ 2231 Mark
$3 \int \sqrt{x} d x=$
  • ✓
    $2 x^{\frac{3}{2}}+k$
  • B
    $4 x^3+k$
  • C
    $3 x^{\frac{3}{2}}+k$
  • D
    $6 x^{\frac{3}{2}}+k$
Answer
Correct option: A.
$2 x^{\frac{3}{2}}+k$
A
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MCQ 2241 Mark
$\int x^{\frac{3}{2}} d x=$
  • A
    $\frac{2}{3} x^{\frac{3}{2}}+k$
  • ✓
    $\frac{2}{5} x^{\frac{5}{2}}+k$
  • C
    $\frac{2}{3} x^{\frac{5}{2}}+k$
  • D
    $\frac{5}{2} x^{\frac{5}{2}}+k$
Answer
Correct option: B.
$\frac{2}{5} x^{\frac{5}{2}}+k$
B
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MCQ 2251 Mark
$\int x e^{2 x} d x=?$
  • A
    $\frac{1}{2} x e^{2 x}+\frac{1}{4} e^{2 x}+C$
  • ✓
    $\frac{1}{2} x e^{2 x}-\frac{1}{4} e^{2 x}+C$
  • C
    $2 x e^{2 x}+4 e^{2 x}+C$
  • D
    इनमें से कोई नहीं
Answer
Correct option: B.
$\frac{1}{2} x e^{2 x}-\frac{1}{4} e^{2 x}+C$
B
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MCQ 2261 Mark
$\int \tan 2 x d x=$
  • A
    $\frac{1}{2} \log |\tan 2 x|+k$
  • ✓
    $\frac{1}{2} \log |\sec 2 x|+k$
  • C
    $2 \log |\sec 2 x|+k$
  • D
    $\log |\sec x|+k$
Answer
Correct option: B.
$\frac{1}{2} \log |\sec 2 x|+k$
B
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MCQ 2281 Mark
$\int\left(\frac{\cos ^2 x+\sin ^2 x}{\operatorname{cosec} x}\right) d x$
  • A
    $k+\operatorname{cosec} x$
  • B
    $k-\sin x$
  • C
    $k+\sin x$
  • ✓
    $k-\cos x$
Answer
Correct option: D.
$k-\cos x$
D
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MCQ 2291 Mark
$\int\left(4 e^{3 x}+1\right) d x=$
  • A
    $4 e^{3 x}+k$
  • ✓
    $\frac{4}{3} e^{3 x}+x+k$
  • C
    $12 e^{3 x}+x+k$
  • D
    $12 e^{3 x}+k$
Answer
Correct option: B.
$\frac{4}{3} e^{3 x}+x+k$
B
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MCQ 2301 Mark
$\int \frac{x^2-1}{x-1} d x=$
  • A
    $x+k$
  • ✓
    $\frac{x^2}{2}+x+k$
  • C
    $\frac{x^3}{3}-\frac{x^2}{2}+k$
  • D
    $\frac{x^2}{2}+2 x+k$
Answer
Correct option: B.
$\frac{x^2}{2}+x+k$
B
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MCQ 2311 Mark
$\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \sin ^3 x d x=$
  • ✓
    $0$
  • B
    1
  • C
    2
  • D
    $\frac{3}{\sqrt{2}}$
Answer
Correct option: A.
$0$
A
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MCQ 2331 Mark
$\int \frac{d x}{x^2+4}=$
  • A
    $\frac{1}{2} \sin ^{-1} \frac{x}{2}+k$
  • B
    $\frac{1}{2} \cos ^{-1} \frac{x}{2}+k$
  • ✓
    $\frac{1}{2} \tan ^{-1} \frac{x}{2}+k$
  • D
    $\tan ^{-1} x+k$
Answer
Correct option: C.
$\frac{1}{2} \tan ^{-1} \frac{x}{2}+k$
C
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MCQ 2341 Mark
$\int \frac{d x}{x^2-1}=$
  • A
    $\sin ^{-1} x+k$
  • ✓
    $\frac{1}{2} \log \left|\frac{x-1}{x+1}\right|+k$
  • C
    $\frac{1}{2} \log \left|\frac{x+1}{x-1}\right|+k$
  • D
    $\sqrt{1-x^2}+k$
Answer
Correct option: B.
$\frac{1}{2} \log \left|\frac{x-1}{x+1}\right|+k$
B
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MCQ 2351 Mark
$\int \frac{1}{\sqrt{1-x^2}} d x=$
  • A
    $\tan ^{-1} x+k$
  • ✓
    $\sin ^{-1} x+k$
  • C
    $\cos ^{-1} x+k$
  • D
    $\sec ^{-1} x+k$
Answer
Correct option: B.
$\sin ^{-1} x+k$
B
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MCQ 2361 Mark
$\int \cos 6 \theta d \theta=$
  • A
    $k+\sin 6 \theta$
  • B
    $k-\sin 6 \theta$
  • C
    $k-6 \sin 6 \theta$
  • ✓
    $k+\frac{1}{6} \sin 6 \theta$
Answer
Correct option: D.
$k+\frac{1}{6} \sin 6 \theta$
D
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MCQ 2391 Mark
$\int_1^4 \sqrt{x} d x=$
  • ✓
    $\frac{14}{3}$
  • B
    $\frac{11}{3}$
  • C
    $\frac{19}{3}$
  • D
    $\frac{22}{3}$
Answer
Correct option: A.
$\frac{14}{3}$
A
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MCQ 2421 Mark
$\int_0^1 x(1-x)^{10} d x=$
  • ✓
    $\frac{1}{132}$
  • B
    $\frac{5}{132}$
  • C
    $\frac{7}{132}$
  • D
    $\frac{45}{244}$
Answer
Correct option: A.
$\frac{1}{132}$
A
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MCQ 2431 Mark
$\int_0^1 \frac{\tan ^{-1} x}{1+x^2} d x=$
  • A
    $\frac{\pi^2}{4}$
  • B
    $\frac{\pi^2}{16}$
  • ✓
    $\frac{\pi^2}{32}$
  • D
    $\frac{\pi^2}{8}$
Answer
Correct option: C.
$\frac{\pi^2}{32}$
C
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MCQ 2441 Mark
$2 \int \frac{1}{1+4 x^2} d x=$
  • A
    $\tan ^{-1} x+k$
  • B
    $\log \left|1+4 x^2\right|+k$
  • ✓
    $\tan ^{-1} 2 x+k$
  • D
    $\sin ^{-1} 2 x+k$
Answer
Correct option: C.
$\tan ^{-1} 2 x+k$
C
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MCQ 2471 Mark
$\int(x+2) d x=$
  • A
    $(x+2)^3+k$
  • B
    $\frac{x^2}{2}+k$
  • ✓
    $\frac{x^2}{2}+2 x+k$
  • D
    $\log (x+2)+k$
Answer
Correct option: C.
$\frac{x^2}{2}+2 x+k$
C
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MCQ 2481 Mark
$\int \sin x \cdot \cos x d x=$
  • A
    $\frac{1}{2} \sin 2 x+k$
  • B
    $\frac{\sin x}{2}+k$
  • ✓
    $\frac{\sin ^2 x}{2}+k$
  • D
    $\frac{\cos ^2 x}{2}+k$
Answer
Correct option: C.
$\frac{\sin ^2 x}{2}+k$
C
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MCQ 2491 Mark
$\int \log x d x=$
  • A
    $\frac{1}{x}+k$
  • B
    $x \log x+k$
  • ✓
    $x \log x-x+k$
  • D
    $x \log x+x+k$
Answer
Correct option: C.
$x \log x-x+k$
C
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MCQ 2511 Mark
$\int \frac{x^4-1}{x^2+1} d x=$
  • A
    $\frac{x^3}{3}+2 x+k$
  • B
    $\frac{x^3}{3}-2 x+k$
  • C
    $\frac{x^3}{3}+x+k$
  • ✓
    $\frac{x^3}{3}-x+k$
Answer
Correct option: D.
$\frac{x^3}{3}-x+k$
D
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MCQ 2521 Mark
$\int \frac{\sec x}{\sec x+\tan x} d x=$
  • A
    $\tan x+\sec x+k$
  • ✓
    $\tan x-\sec x+k$
  • C
    $\sec x+k$
  • D
    $\tan x+k$
Answer
Correct option: B.
$\tan x-\sec x+k$
B
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MCQ 2541 Mark
$\int \frac{d x}{x-1}=$
  • A
    $\log |x+1|+k$
  • B
    $-\log |1+x|+k$
  • ✓
    $\log |x-1|+k$
  • D
    $\log x+k$
Answer
Correct option: C.
$\log |x-1|+k$
C
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MCQ 2551 Mark
$\int \frac{1-\cos 2 x}{1+\cos 2 x} d x$
  • A
    $\tan x+x+k$
  • ✓
    $\tan x-x+k$
  • C
    $x-\tan ^2 x+k$
  • D
    $\tan \frac{x}{2}+k$
Answer
Correct option: B.
$\tan x-x+k$
B
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MCQ 2571 Mark
$\int_0^{\frac{\pi}{2}} \log \cot \theta d \theta=$
  • A
    $\frac{\pi}{2} \log 2$
  • B
    $\frac{\pi}{4} \log 2$
  • C
    $2 \pi \log 2$
  • ✓
    $0$
Answer
Correct option: D.
$0$
D
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MCQ 2581 Mark
$\int_0^{\frac{\pi}{2}} \frac{\cos \theta}{\cos \theta+\sin \theta} d \theta$
  • A
    $\pi$
  • B
    $\frac{\pi}{2}$
  • C
    $\frac{\pi}{3}$
  • ✓
    $\frac{\pi}{4}$
Answer
Correct option: D.
$\frac{\pi}{4}$
D
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MCQ 2601 Mark
$\int_0^2\left(x^2+1\right) d x=$
  • A
    $\frac{8}{3}$
  • ✓
    $\frac{14}{3}$
  • C
    $\frac{13}{3}$
  • D
    $\frac{1}{3}$
Answer
Correct option: B.
$\frac{14}{3}$
B
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MCQ 2631 Mark
$\int x^2 \sin x^3 d x=$
  • ✓
    $-\frac{1}{3} \cos x^3+k$
  • B
    $\frac{1}{3} \cos x^3+k$
  • C
    $\frac{1}{3} \sin x^3+k$
  • D
    $-\frac{1}{3} \sin x^3+k$
Answer
Correct option: A.
$-\frac{1}{3} \cos x^3+k$
A
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MCQ 2641 Mark
$\int \tan x d x=$
  • ✓
    $\log |\sec x|+k$
  • B
    $\log |\sin x|+k$
  • C
    $\log |\cos x|+k$
  • D
    $\log |\operatorname{cosec} x|+k$
Answer
Correct option: A.
$\log |\sec x|+k$
A
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MCQ 2651 Mark
$\int \sqrt{1-\sin 2 x} d x=$
  • ✓
    $\sin x+\cos x+k$
  • B
    $\sin x-\cos x+k$
  • C
    $\cos x-\sin x+k$
  • D
    $\tan x-\cot x+k$
Answer
Correct option: A.
$\sin x+\cos x+k$
A
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MCQ 2661 Mark
$\int \sec ^2(3 x+5) d x=$
  • ✓
    $\frac{1}{3} \tan (3 x+5)+k$
  • B
    $-\frac{1}{3} \tan (3 x+5)+k$
  • C
    $\frac{1}{5} \tan (3 x+5)+k$
  • D
    $-\frac{1}{5} \tan (3 x+5)+k$
Answer
Correct option: A.
$\frac{1}{3} \tan (3 x+5)+k$
A
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MCQ 2671 Mark
$\int \operatorname{cosec} x d x=$
  • ✓
    $\log \left|\tan \frac{x}{2}\right|+k$
  • B
    $\log \left|\cot \frac{x}{2}\right|+k$
  • C
    $\log \left|\sin \frac{x}{2}\right|+k$
  • D
    $\log \left|\cos \frac{x}{2}\right|+k$
Answer
Correct option: A.
$\log \left|\tan \frac{x}{2}\right|+k$
A
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MCQ 2681 Mark
$\int \frac{\sin x+\cos x}{\sqrt{1+\sin 2 x}} d x=$
  • A
    $\log x+C$
  • ✓
    $x+C$
  • C
    $\log (\sin x+\cos x)+C$
  • D
    $\log \sin (\cos x)+C$
Answer
Correct option: B.
$x+C$
B
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MCQ 2691 Mark
$\int \frac{\sec ^2(\log x)}{x} d x=?$
  • A
    $\log (\tan x)+C$
  • B
    $-\log (\tan x)+C$
  • C
    $\tan (\tan x)+C$
  • ✓
    $\tan (\log x)+C$
Answer
Correct option: D.
$\tan (\log x)+C$
D
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MCQ 2701 Mark
$\int e^x(\cot x+\log \sin x) d x=$
  • ✓
    $e^x \log \sin x+k$
  • B
    $e^x \cot x+k$
  • C
    $e^x \sin x+k$
  • D
    $e^* \cos x+k$
Answer
Correct option: A.
$e^x \log \sin x+k$
A
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MCQ 2721 Mark
$\int_4^9 \sqrt{x} d x=$
  • ✓
    $\frac{38}{3}$
  • B
    $\frac{35}{3}$
  • C
    $\frac{36}{7}$
  • D
    $\frac{45}{3}$
Answer
Correct option: A.
$\frac{38}{3}$
A
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MCQ 2731 Mark
$\int_0^{\frac{\pi}{4}} \tan ^2 x d x=$
  • A
    $\frac{\pi}{4}$
  • ✓
    $1-\frac{\pi}{4}$
  • C
    $\frac{\pi}{4}-1$
  • D
    $1+\frac{\pi}{4}$
Answer
Correct option: B.
$1-\frac{\pi}{4}$
B
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MCQ 2741 Mark
$\int_0^{\frac{\pi}{2}} \frac{\sin x}{\sin x+\cos x} d x=$
  • ✓
    $\frac{\pi}{4}$
  • B
    $\frac{\pi}{2}$
  • C
    $-\frac{\pi}{4}$
  • D
    $-\frac{\pi}{2}$
Answer
Correct option: A.
$\frac{\pi}{4}$
A
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MCQ 2751 Mark
$\int_0^2 e^{\frac{x}{2}} d x=$
  • ✓
    $2(e-1)$
  • B
    $2(e+1)$
  • C
    $2(1-e)$
  • D
    $2\left(e^2-1\right)$
Answer
Correct option: A.
$2(e-1)$
A
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MCQ 2771 Mark
$\int \operatorname{cosec}^2 x d x=$
  • A
    $\tan x+C$
  • ✓
    $-\cot x+C$
  • C
    $2 \operatorname{cosec} x+C$
  • D
    $-2 \operatorname{cosec} x+C$
Answer
Correct option: B.
$-\cot x+C$
B
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MCQ 2781 Mark
$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^9 x d x$ बराबर है
  • A
    -1
  • ✓
    $0$
  • C
    1
  • D
    $\frac{\pi}{2}$
Answer
Correct option: B.
$0$
B
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MCQ 2791 Mark
$\int \frac{d x}{x^2+16}=$
  • A
    $\frac{1}{16} \tan ^{-1} \frac{x}{16}+k$
  • ✓
    $\frac{1}{4} \tan ^{-1} \frac{x}{4}+k$
  • C
    $\frac{1}{4} \tan ^{-1} \frac{4}{x}+k$
  • D
    $\frac{1}{4} \tan ^{-1} \frac{16}{x^2}+k$
Answer
Correct option: B.
$\frac{1}{4} \tan ^{-1} \frac{x}{4}+k$
B
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MCQ 2801 Mark
$\int_2^1 \frac{d x}{x}=$
  • A
    $\log \frac{2}{3}$
  • B
    $\log \frac{3}{2}$
  • ✓
    $\log \frac{1}{2}$
  • D
    $\log \frac{x}{2}$
Answer
Correct option: C.
$\log \frac{1}{2}$
C
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MCQ 2811 Mark
$\int_0^{\frac{\pi}{2}} \sin ^3 x \cos x d x=$
  • A
    $\frac{1}{3}$
  • ✓
    $\frac{1}{4}$
  • C
    $\frac{1}{2}$
  • D
    $\frac{\pi}{2}$
Answer
Correct option: B.
$\frac{1}{4}$
B
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MCQ 2821 Mark
$\int_0^1 \frac{d x}{1+x^2}=$
  • A
    $-\frac{\pi}{4}$
  • ✓
    $\frac{\pi}{4}$
  • C
    $\frac{\pi}{2}$
  • D
    $-\frac{\pi}{2}$
Answer
Correct option: B.
$\frac{\pi}{4}$
B
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MCQ 2831 Mark
$\int \cot ^2 x d x=$
  • A
    $\cot x-x+C$
  • ✓
    $-\cot x-x+C$
  • C
    $-2 \cot x \operatorname{cosec}^2 x+C$
  • D
    $-\operatorname{cosec}^2 x+C$
Answer
Correct option: B.
$-\cot x-x+C$
B
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MCQ 2841 Mark
$\operatorname{Lt}_{n \rightarrow \infty} \sum_{r=1}^n \frac{r^3}{r^4+n^4}$ का मान है
  • A
    $\log 2$
  • ✓
    $\frac{1}{4} \log 2$
  • C
    $-\frac{1}{4} \log 2$
  • D
    $4 \log 2$
Answer
Correct option: B.
$\frac{1}{4} \log 2$
B
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MCQ 2851 Mark
$\int \frac{e^{2 x}-1}{e^{2 x}+1} d x=$
  • A
    $\log \left(e^{2 x}+1\right)+C$
  • ✓
    $\log \left(e^x+e^{-x}\right)+C$
  • C
    $\log \left(e^x-e^{-x}\right)+C$
  • D
    $e^{2 x}+C$
Answer
Correct option: B.
$\log \left(e^x+e^{-x}\right)+C$
B
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MCQ 2861 Mark
$\int \cos x^0 d x=$
  • A
    $\sin x^0 d x$
  • B
    $\frac{\pi}{180} \sin x^0+C$
  • ✓
    $\frac{180}{\pi} \sin x^0+C$
  • D
    $\frac{\pi x}{180} \sin x^0+C$
Answer
Correct option: C.
$\frac{180}{\pi} \sin x^0+C$
C
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MCQ 2891 Mark
$\int_0^{\frac{x}{2}} \log \tan x d x=$
  • A
    $\frac{\pi}{2} \log \frac{1}{2}$
  • B
    $\frac{\pi}{2} \log 2$
  • C
    1
  • ✓
    $0$
Answer
Correct option: D.
$0$
D
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MCQ 2901 Mark
$\int_\alpha^\beta \phi(x) d x+\int_\beta^a \phi(x) d x=$
  • A
    1
  • B
    $2 \int_\alpha^\beta \phi(x) d x$
  • C
    $-2 \int_\beta^\alpha \phi(x) d x$
  • ✓
    $0$
Answer
Correct option: D.
$0$
D
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MCQ 2911 Mark
यदि $f(-x)=-f(x)$, then $\int_{-a}^a f(x) d x=$
  • A
    $2 \int_0^a f(x) d x$
  • ✓
    $0$
  • C
    1
  • D
    $-1$
Answer
Correct option: B.
$0$
B
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MCQ 2921 Mark
$\lim _{x \rightarrow \infty}\left[\frac{e^{1 / n}+e^{2 / n}+e^{3 / n} \ldots \ldots+e^{n / n}}{n}\right]$ का मान है
  • A
    $1-e$
  • ✓
    $e-1$
  • C
    $e$
  • D
    1
Answer
Correct option: B.
$e-1$
B
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MCQ 2931 Mark
$\int \frac{x e^x}{(x+1)^2} d x=$
  • A
    $\frac{e^x}{(x+1)^2}+C$
  • B
    $-\frac{e^x}{x+1}+C$
  • ✓
    $\frac{e^x}{x+1}+C$
  • D
    $-\frac{e^x}{(x+1)^2}+C$
Answer
Correct option: C.
$\frac{e^x}{x+1}+C$
C
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MCQ 2941 Mark
$\int \frac{d x}{x+\sqrt{x}} d x$
  • A
    $\log x+\log (1+\sqrt{x})+C$
  • ✓
    $2 \log (\sqrt{x}+1)+C$
  • C
    $\log (1+\sqrt{x})+C$
  • D
    $\log \sqrt{x}+C$
Answer
Correct option: B.
$2 \log (\sqrt{x}+1)+C$
B
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MCQ 2981 Mark
$\int_0^{\frac{\pi}{4}} \tan ^2 \theta d \theta=$
  • ✓
    $1-\frac{\pi}{4}$
  • B
    $1+\frac{\pi}{4}$
  • C
    $-1-\frac{\pi}{4}$
  • D
    $\frac{\pi}{4}$
Answer
Correct option: A.
$1-\frac{\pi}{4}$
A
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MCQ 2991 Mark
$\int_0^{\frac{\pi}{2}} \sin ^2 x d x=$
  • ✓
    $\frac{\pi}{4}$
  • B
    $-\frac{\pi}{4}$
  • C
    $\frac{\pi}{2}$
  • D
    $-\frac{\pi}{2}$
Answer
Correct option: A.
$\frac{\pi}{4}$
A
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MCQ 3001 Mark
$\int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=$
  • ✓
    $\frac{\pi}{4}$
  • B
    $\frac{\pi}{2}$
  • C
    $-\frac{\pi}{2}$
  • D
    $-\frac{\pi}{4}$
Answer
Correct option: A.
$\frac{\pi}{4}$
A
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MCQ 3011 Mark
$\int_0^{\frac{\pi}{2}} e^x(\sin x+\cos x) d x=$
  • ✓
    $e^{\frac{\pi}{2}}$
  • B
    1
  • C
    $e^{\frac{\pi}{4}}$
  • D
    $e^\pi$
Answer
Correct option: A.
$e^{\frac{\pi}{2}}$
A
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MCQ 3021 Mark
$\int \frac{d x}{x}=$.................
  • A
    $x+k$
  • B
    $\frac{1}{x^2}+k$
  • C
    $-\frac{1}{x^2}+k$
  • ✓
    $\log x+k$
Answer
Correct option: D.
$\log x+k$
D
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MCQ 3031 Mark
$\int x^5 d x=$...............
  • ✓
    $\frac{x^6}{6}+k$
  • B
    $\frac{x^5}{5}+k$
  • C
    $\frac{x^7}{7}+k$
  • D
    $\frac{x^8}{8}+k$
Answer
Correct option: A.
$\frac{x^6}{6}+k$
A
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MCQ 3041 Mark
$\int_a^b x^3 d x=\ldots \ldots \ldots$
  • A
    $\frac{b^3-a^3}{3}$
  • ✓
    $\frac{b^4-a^4}{4}$
  • C
    $\frac{b^2-a^2}{2}$
  • D
    $0$
Answer
Correct option: B.
$\frac{b^4-a^4}{4}$
B
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MCQ 3061 Mark
$\int x^n d x, n \neq 0=$
  • A
    $\frac{x^{n-1}}{n-1}+C$
  • ✓
    $\frac{x^{n+1}}{n+1}+C$
  • C
    $x^{n+1}+C$
  • D
    $x^{n-1}+C$
Answer
Correct option: B.
$\frac{x^{n+1}}{n+1}+C$
B
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MCQ 3071 Mark
$\int \sec x d x=$
  • A
    $\log \sec x+C$
  • ✓
    $\log (\sec x+\tan x)+C$
  • C
    $\log (\sec x-\tan x)+C$
  • D
    $\sec x \tan x$
Answer
Correct option: B.
$\log (\sec x+\tan x)+C$
B
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MCQ 3081 Mark
$\int \frac{d x}{1+x^2}=$
  • A
    $\cot ^{-1} x+C$
  • ✓
    $\tan ^{-1} x+C$
  • C
    $\sec ^{-1} x+C$
  • D
    $\operatorname{cosec}^{-1} x+C$
Answer
Correct option: B.
$\tan ^{-1} x+C$
B
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MCQ 3091 Mark
$\int \frac{d x}{x^2+a^2}$
  • A
    $\tan ^{-1} \frac{x}{a}+C$
  • B
    $\frac{1}{a} \tan ^{-1} \frac{a}{x}+C$
  • ✓
    $\frac{1}{a} \tan ^{-1} \frac{x}{a}+C$
  • D
    $\tan ^{-1} \frac{a}{x}+C$
Answer
Correct option: C.
$\frac{1}{a} \tan ^{-1} \frac{x}{a}+C$
C
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MCQ 3101 Mark
$\int\left(\sin ^{-1} x+\cos ^{-1} x\right) d x=$
  • A
    $\frac{\pi}{2}+C$
  • ✓
    $\frac{\pi}{2} x+C$
  • C
    $\frac{2}{\sqrt{1-x^2}}+C$
  • D
    $x+C$
Answer
Correct option: B.
$\frac{\pi}{2} x+C$
B
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MCQ 3111 Mark
$\int_0^a f(x) d x=$
  • A
    $\int_0^a f(a+x) d x$
  • ✓
    $\int_0^a f(a-x) d x$
  • C
    $\int_0^a f(x-a) d x$
  • D
    $\int_0^a f(a) d x$
Answer
Correct option: B.
$\int_0^a f(a-x) d x$
B
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MCQ 3121 Mark
$\int_a^b x^2 d x=$
  • ✓
    $\frac{b^3-a^3}{3}$
  • B
    $\frac{a^3-b^3}{3}$
  • C
    $\frac{a^2-b^2}{2}$
  • D
    $\frac{b^2-a^2}{2}$
Answer
Correct option: A.
$\frac{b^3-a^3}{3}$
A
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MCQ 3141 Mark
$\int x^8 d x=?$
  • A
    $8 x^7+k$
  • B
    $\frac{x^8}{8}+k$
  • C
    $x^9+k$
  • ✓
    $\frac{x^9}{9}+k$
Answer
Correct option: D.
$\frac{x^9}{9}+k$
D
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MCQ 3151 Mark
$\int \frac{d x}{1-\sin x}=$
  • A
    $\tan x-\sec x+k$
  • ✓
    $\tan x+\sec x+k$
  • C
    $\tan ^2 x+\sec ^2 x+k$
  • D
    $2(\tan x-\sec x)+k$
Answer
Correct option: B.
$\tan x+\sec x+k$
B
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MCQ 3161 Mark
$\int \frac{-1}{1+x^2} d x=?$
  • A
    $\tan ^{-1} x+k$
  • B
    $\sec ^{-1} x+k$
  • C
    $\operatorname{cosec}^{-1} x+k$
  • ✓
    $\cot ^{-1} x+k$
Answer
Correct option: D.
$\cot ^{-1} x+k$
D
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MCQ 3171 Mark
यदि $x > a, \int \frac{d x}{x^2-a^2}=$
  • ✓
    $\frac{1}{2 a} \log \frac{x-a}{x+a}$
  • B
    $\frac{1}{2 a} \log \frac{x+a}{x-a}+k$
  • C
    $\frac{1}{a} \log \left(x^2-a^2\right)+k$
  • D
    $\log \left(x+\sqrt{x^2-a^2}\right)+k$
Answer
Correct option: A.
$\frac{1}{2 a} \log \frac{x-a}{x+a}$
A
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MCQ 3181 Mark
$\int \frac{d x}{\sqrt{x}}=$
  • A
    $\sqrt{x}+k$
  • ✓
    $2 \sqrt{x}+k$
  • C
    $x+k$
  • D
    $\frac{2}{3} x^{\frac{3}{2}}+k$
Answer
Correct option: B.
$2 \sqrt{x}+k$
B
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MCQ 3191 Mark
$\int \frac{d x}{1+\cos x}=$
  • ✓
    $\tan \frac{x}{2}+k$
  • B
    $\frac{1}{2} \tan \frac{x}{2}+k$
  • C
    $2 \tan \frac{x}{2}+k$
  • D
    $\tan ^2 \frac{x}{2}+k$
Answer
Correct option: A.
$\tan \frac{x}{2}+k$
A
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MCQ 3201 Mark
$\int_a^b x^5 d x=$
  • A
    $b^5-a^5$
  • ✓
    $\frac{b^6-a^6}{6}$
  • C
    $\frac{a^6-b^6}{6}$
  • D
    $a^5-b^5$
Answer
Correct option: B.
$\frac{b^6-a^6}{6}$
B
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MCQ 3221 Mark
$\int\left[\frac{d}{d x}\left(\log _{ e } x\right)\right] d x=\ldots \ldots \ldots+k$ जहाँ k अचर है।
  • ✓
    $\log _e x$
  • B
    $\log x+\log (1+\sqrt{x})$
  • C
    $\log (1+\sqrt{x})+k$
  • D
    $\log _e \sqrt{x}$
Answer
Correct option: A.
$\log _e x$
A
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MCQ 3231 Mark
$\int_0^{\frac{\pi}{4}} \log (1+\tan x) d x=$
  • A
    $\frac{\pi}{4}$
  • B
    $\frac{\pi}{8}$
  • C
    $\frac{\pi}{4} \log 2$
  • ✓
    $\frac{\pi}{8} \log 2$
Answer
Correct option: D.
$\frac{\pi}{8} \log 2$
D
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MCQ 3241 Mark
$\int_0^{\frac{\pi}{2}} \sin 2 x \cdot \log \tan x d x=$
  • A
    2
  • B
    $\frac{\pi}{2}$
  • ✓
    $0$
  • D
    1
Answer
Correct option: C.
$0$
C
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MCQ 3251 Mark
$\int_0^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=$ ?
  • ✓
    $\frac{\pi}{4}$
  • B
    $\frac{-\pi}{4}$
  • C
    $0$
  • D
    $\frac{\pi}{2}$
Answer
Correct option: A.
$\frac{\pi}{4}$
A
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MCQ 3261 Mark
$\int e^x\left(\frac{1}{x}-\frac{1}{x^2}\right) d x=$
  • ✓
    $\frac{e^x}{x}+k$
  • B
    $\frac{e^x}{x^2}+k$
  • C
    $\frac{e^{2 x}}{x^2}+k$
  • D
    $\frac{e^{x^2}}{x^2}+k$
Answer
Correct option: A.
$\frac{e^x}{x}+k$
A
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MCQ 3271 Mark
$\int e^x(\tan x+\log \sec x) d x=$
  • ✓
    $e^x \log \sec x+k$
  • B
    $e^x \tan x+k$
  • C
    $e^x \log \cos x+k$
  • D
    $e^x \sin x+k$
Answer
Correct option: A.
$e^x \log \sec x+k$
A
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MCQ 3281 Mark
$\int e^x \frac{1+\sin x}{1+\cos x} d x=$
  • ✓
    $e^x \tan \frac{x}{2}+k$
  • B
    $e^x \sin \frac{x}{2}+k$
  • C
    $e^x \cos \frac{x}{2}+k$
  • D
    $e^x \cot \frac{x}{2}+k$
Answer
Correct option: A.
$e^x \tan \frac{x}{2}+k$
A
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MCQ 3291 Mark
$\int \frac{(1+\log x)^2}{x} d x=$
  • ✓
    $\frac{1}{3}(1+\log x)^3+C$
  • B
    $\frac{1}{2}(1+\log x)^2+C$
  • C
    $\log \{\log (1+x)\}+C$
  • D
    इनमें से कोई नहीं
Answer
Correct option: A.
$\frac{1}{3}(1+\log x)^3+C$
A
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MCQ 3311 Mark
$\int_0^1 \frac{\left(\tan ^{-1} x\right)^2}{1+x^2} d x=$
  • A
    1
  • B
    $\frac{\pi^3}{64}$
  • ✓
    $\frac{\pi^3}{192}$
  • D
    इनमें से कोई नहीं
Answer
Correct option: C.
$\frac{\pi^3}{192}$
C
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MCQ 3321 Mark
$\int \frac{e^{\tan ^{-1} x}}{\left(1+x^2\right)} d x=?$
  • A
    $\frac{e^{\tan ^{-1} x}}{x}+C$
  • ✓
    $e^{\tan ^{-1} x}+C$
  • C
    $e^x \tan ^{-1} x+C$
  • D
    इनमें से कोई नहीं
Answer
Correct option: B.
$e^{\tan ^{-1} x}+C$
B
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MCQ 3331 Mark
यदि $\int f(x) d x=f(x)+C$ तो
  • A
    $f(x)=x$
  • B
    $f(x)=x^2$
  • C
    $f(x)=\frac{1}{x}$
  • ✓
    $f(x)=e^x$
Answer
Correct option: D.
$f(x)=e^x$
D
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MCQ 3341 Mark
यदि $\int_0^k \frac{\cos x}{1+\sin ^2 x} d x=\frac{\pi}{4}$, तब $k=$
  • A
    1
  • B
    $\frac{\pi}{4}$
  • ✓
    $\frac{\pi}{2}$
  • D
    $\pi$
Answer
Correct option: C.
$\frac{\pi}{2}$
C
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MCQ 3351 Mark
$\operatorname{Lt}_{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{\sqrt{n r}}=$
  • ✓
    2
  • B
    1
  • C
    $0$
  • D
    3
Answer
Correct option: A.
2
A
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MCQ 3361 Mark
$\int x^{\frac{5}{3}} d x=?$
  • A
    $\frac{3}{5} x^{\frac{2}{3}}+C$
  • B
    $\frac{8}{3} x^{\frac{8}{3}}+C$
  • ✓
    $\frac{3}{8} x^{\frac{8}{3}}+C$
  • D
    $\frac{5}{3} x^{\frac{8}{3}}+C$
Answer
Correct option: C.
$\frac{3}{8} x^{\frac{8}{3}}+C$
C
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MCQ 3371 Mark
$\int x^6 d x=?$
  • A
    $7 x^7+C$
  • ✓
    $\frac{x^7}{7}+C$
  • C
    $6 x^5+C$
  • D
    $6 x^7+C$
Answer
Correct option: B.
$\frac{x^7}{7}+C$
B
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MCQ 3381 Mark
$\int(\tan x+\cot x) d x=$
  • ✓
    $\log (C \tan x)$
  • B
    $\log (\sin x+\cos x)+C$
  • C
    $\log (C x)$
  • D
    इनमें से कोई नहीं
Answer
Correct option: A.
$\log (C \tan x)$
A
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MCQ 3391 Mark
$\int_{-\pi}^\pi x \cos ^3 x d x=$
  • A
    $2 \int_0^\pi x \cos ^3 x d x$
  • B
    $\frac{2 \pi}{3}$
  • ✓
    $0$
  • D
    $\frac{\pi}{3}$
Answer
Correct option: C.
$0$
C
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MCQ 3411 Mark
$\int \frac{\tan (\log x)}{x} d x=$
  • A
    $\log |\cos (\log x)|+C$
  • ✓
    $-|\log | \cos (\log x) \mid+C$
  • C
    $x \tan (\log x)+C$
  • D
    $\log |\tan x|+C$
Answer
Correct option: B.
$-|\log | \cos (\log x) \mid+C$
B
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MCQ 3421 Mark
$\int \frac{\tan ^{-1} x}{1+x^2} d x=$
  • ✓
    $\frac{\left(\tan ^{-1} x\right)^2}{2}+C$
  • B
    $\log \left|\tan ^{-1} x\right|+C$
  • C
    $x \tan ^{-1} x+C$
  • D
    $x \sin ^{-1} x+C$
Answer
Correct option: A.
$\frac{\left(\tan ^{-1} x\right)^2}{2}+C$
A
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MCQ 3431 Mark
$\int \frac{\sin x}{\left(1+\cos ^2 x\right)} d x=?$
  • ✓
    $-\tan ^{-1}(\cos x)+C$
  • B
    $\cot ^{-1}(\cos x)+C$
  • C
    $-\cot ^{-1}(\cos x)+C$
  • D
    $\tan ^{-1}(\cos x)+C$
Answer
Correct option: A.
$-\tan ^{-1}(\cos x)+C$
A
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MCQ 3441 Mark
$\int \frac{\sin x}{1-\sin ^2 x} d x=$
  • A
    $-\sec x+C$
  • B
    $\log \left|\frac{1+\sin x}{1-\sin x}\right|+C$
  • ✓
    $\sec x+C$
  • D
    $\cos x+C$
Answer
Correct option: C.
$\sec x+C$
C
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MCQ 3451 Mark
$\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x=?$
  • A
    $2 \cos \sqrt{x}+C$
  • ✓
    $-2 \cos \sqrt{x}+C$
  • C
    $-\frac{\cos \sqrt{x}}{2}+C$
  • D
    $\frac{\cos \sqrt{x}}{2}+C$
Answer
Correct option: B.
$-2 \cos \sqrt{x}+C$
B
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MCQ 3461 Mark
$\int \frac{\sin ^2 x}{\cos ^2 x} d x=$
  • ✓
    $\tan x-x+C$
  • B
    $-\tan x-x+C$
  • C
    $\cot x-x+C$
  • D
    $\tan x+x+C$
Answer
Correct option: A.
$\tan x-x+C$
A
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MCQ 3471 Mark
$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{\sin 2 x}$ का मान क्या है?
  • A
    $\frac{1}{2} \log (-1)$
  • B
    $\log (-1)$
  • C
    $\log 3$
  • ✓
    $\log \sqrt{3}$
Answer
Correct option: D.
$\log \sqrt{3}$
D
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MCQ 3481 Mark
$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot x d x=$
  • A
    $2 \log 2$
  • B
    $\frac{\pi}{2} \log 2$
  • ✓
    $\log \sqrt{2}$
  • D
    $\log 2$
Answer
Correct option: C.
$\log \sqrt{2}$
C
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MCQ 3491 Mark
$\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}}|\cos x| d x$
  • A
    $0$
  • B
    $\sqrt{2}$
  • C
    $2+\sqrt{2}$
  • ✓
    $2-\sqrt{2}$
Answer
Correct option: D.
$2-\sqrt{2}$
D
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MCQ 3501 Mark
$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\frac{1}{2}(1-\cos 2 x)} d x=$
  • A
    $0$
  • B
    1
  • ✓
    2
  • D
    -2
Answer
Correct option: C.
2
C
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MCQ 3511 Mark
$\int \frac{\log x}{x} d x=?$
  • ✓
    $\frac{1}{2}(\log x)^2+C$
  • B
    $-\frac{1}{2}(\log x)^2+C$
  • C
    $\frac{2}{x^2}+C$
  • D
    $\frac{-2}{x^2}+C$
Answer
Correct option: A.
$\frac{1}{2}(\log x)^2+C$
A
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MCQ 3521 Mark
$\int \frac{e^x}{x}(1+x \log x) d x=$
  • A
    $x e^x+K$
  • B
    $\frac{e^x}{x}+K$
  • ✓
    $e^x \log x+K$
  • D
    इनमें से कोई नहीं
Answer
Correct option: C.
$e^x \log x+K$
C
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MCQ 3531 Mark
$\int \frac{e^x}{\sqrt{1+e^x}} d x=$
  • ✓
    $2 \sqrt{1+e^x}+C$
  • B
    $\sqrt{1+e^x}+C$
  • C
    $\frac{1}{\sqrt{1+e^x}}+C$
  • D
    $\frac{2}{\sqrt{1+e^x}}+C$
Answer
Correct option: A.
$2 \sqrt{1+e^x}+C$
A
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MCQ 3541 Mark
$\int \frac{e^x-e^{-x}}{e^x+e^{-x}} d x=$
  • A
    $\log \left|e^x-e^{-x}\right|+C$
  • B
    $\log \cos x+C$
  • C
    $\log \sin x+C$
  • ✓
    $\log \left|e^x+e^{-x}\right|+C$
Answer
Correct option: D.
$\log \left|e^x+e^{-x}\right|+C$
D
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MCQ 3551 Mark
$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x=?$
  • A
    $e^{\sqrt{x}}+C$
  • B
    $\frac{1}{2} e^{\sqrt{x}}+C$
  • ✓
    $2 e^{\sqrt{x}}+C$
  • D
    इनमें से कोई नहीं
Answer
Correct option: C.
$2 e^{\sqrt{x}}+C$
C
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MCQ 3561 Mark
$\int \frac{d x}{x^2-256}=$
  • ✓
    $\frac{1}{32} \log \left|\frac{x-16}{x+16}\right|+k$
  • B
    $\frac{1}{32} \log \left|x^2-256\right|+k$
  • C
    $\frac{1}{16} \log \left|\frac{x-16}{x+16}\right|+k$
  • D
    $\sin ^{-1}\left(\frac{x}{16}\right)+k$
Answer
Correct option: A.
$\frac{1}{32} \log \left|\frac{x-16}{x+16}\right|+k$
A
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MCQ 3571 Mark
$\int \frac{3}{x} d x=$
  • A
    $k+3 x^2$
  • B
    $k-\frac{3}{x^2}$
  • C
    $3 x+k$
  • ✓
    $k+3 \log |x|$
Answer
Correct option: D.
$k+3 \log |x|$
D
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MCQ 3581 Mark
$\int \frac{1-x^2}{1+x^2} d x=$
  • A
    $x+2 \tan ^{-1} x+C$
  • B
    $-x+\tan ^{-1} x+C$
  • C
    $-x+3 \tan ^{-1} x+C$
  • ✓
    $-x+2 \tan ^{-1} x+C$
Answer
Correct option: D.
$-x+2 \tan ^{-1} x+C$
D
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MCQ 3591 Mark
$\int \frac{1}{\sqrt[3]{x}} d x=?$
  • ✓
    $\frac{3}{2} x^{\frac{2}{3}}+C$
  • B
    $\frac{3}{2 x^{\frac{2}{3}}}+C$
  • C
    $\frac{2}{3 x^{\frac{2}{3}}}+C$
  • D
    $\frac{2}{3} x^{\frac{3}{2}}+C$
Answer
Correct option: A.
$\frac{3}{2} x^{\frac{2}{3}}+C$
A
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MCQ 3601 Mark
$\int e^x \sin \left(e^x\right) d x=$
  • A
    $\cos \left(e^x\right)+C$
  • ✓
    $-\cos \left(e^x\right)+C$
  • C
    $\sin \left(e^x\right)+C$
  • D
    $\tan \left(e^x\right)+C$
Answer
Correct option: B.
$-\cos \left(e^x\right)+C$
B
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MCQ 3611 Mark
$\int e^{3 x} d x=$
  • A
    $e^{3 x}+k$
  • B
    $k+3 e^{3 k}$
  • ✓
    $\frac{e^{3 x}}{3}+k$
  • D
    $\frac{e^{3 x}}{4}+k$
Answer
Correct option: C.
$\frac{e^{3 x}}{3}+k$
C
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MCQ 3621 Mark
$\int e^{2 \log \cot x} d x=$
  • A
    $-\cot x+C$
  • B
    $-\cot x+x+C$
  • ✓
    $-\cot x-x+C$
  • D
    $\tan x+C$
Answer
Correct option: C.
$-\cot x-x+C$
C
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MCQ 3631 Mark
$\int_a^b f(x) d x+\int_b^c f(x) d x$ किसके समान है?
  • ✓
    $\int_a^c f(x) d x$
  • B
    $\int_e^a f(x) d x$
  • C
    $\int_{a+b}^{b+c} f(x) d x$
  • D
    इनमें से कोई नहीं
Answer
Correct option: A.
$\int_a^c f(x) d x$
A
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MCQ 3651 Mark
$\int 3^x d x=?$
  • A
    $3^x(\log 3)+C$
  • B
    $3^x+C$
  • ✓
    $\frac{3^x}{\log 3}+C$
  • D
    $\frac{\log 3}{3^x}+C$
Answer
Correct option: C.
$\frac{3^x}{\log 3}+C$
C
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MCQ 3661 Mark
$\int_{-2}^2 \frac{d x}{4+x^2}=$
  • ✓
    $\frac{\pi}{4}$
  • B
    $\pi$
  • C
    $\frac{\pi}{2}$
  • D
    $-\frac{\pi}{4}$
Answer
Correct option: A.
$\frac{\pi}{4}$
A
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MCQ 3671 Mark
$\int_1^e \frac{1+\log x}{x} d x$ का मान क्या है?
  • ✓
    $\frac{3}{2}$
  • B
    $\frac{1}{2}$
  • C
    $e$
  • D
    $\frac{1}{e}$
Answer
Correct option: A.
$\frac{3}{2}$
A
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MCQ 3681 Mark
$\int_1^2 x^{-2} e^{\frac{1}{x}} d x=$
  • ✓
    $e-\sqrt{e}$
  • B
    $e+\sqrt{e}$
  • C
    $\sqrt{e}-1$
  • D
    $\sqrt{e}-2$
Answer
Correct option: A.
$e-\sqrt{e}$
A
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MCQ 3701 Mark
$\int_{-1}^1 \sin ^3 x \cdot \cos ^2 x \cdot d x$ का मान क्या है?
  • ✓
    $0$
  • B
    1
  • C
    $\frac{1}{2}$
  • D
    2
Answer
Correct option: A.
$0$
A
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MCQ 3721 Mark
$\int_0^\pi x \sin ^2 x d x=$
  • A
    $0$
  • B
    $\pi$
  • C
    $\frac{\pi^2}{2}$
  • ✓
    $\frac{\pi^2}{4}$
Answer
Correct option: D.
$\frac{\pi^2}{4}$
D
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MCQ 3731 Mark
$\int_0^{\frac{\pi}{4}} \tan \theta \sec ^2 \theta d \theta=$
  • A
    2
  • B
    3
  • ✓
    $\frac{1}{2}$
  • D
    1
Answer
Correct option: C.
$\frac{1}{2}$
C
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MCQ 3741 Mark
$\int_0^{\frac{\pi}{2}} \sin ^3 \theta \cos \theta d \theta=$
  • ✓
    $\frac{1}{4}$
  • B
    4
  • C
    2
  • D
    $\frac{1}{2}$
Answer
Correct option: A.
$\frac{1}{4}$
A
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MCQ 3751 Mark
$\int_0^{\frac{\pi}{2}} \sin 2 x \log \cot x d x=$
  • ✓
    $0$
  • B
    $\frac{\pi}{2}$
  • C
    $\pi$
  • D
    $\frac{\pi}{4}$
Answer
Correct option: A.
$0$
A
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MCQ 3761 Mark
$\int_0^{\frac{\pi}{2}} \frac{\sqrt{\tan x}}{\sqrt{\tan x}+\sqrt{\cot x}} d x$ का मान क्या है?
  • A
    $\frac{\pi}{2}$
  • ✓
    $\frac{\pi}{4}$
  • C
    $\frac{\pi}{8}$
  • D
    इनमें से कोई नहीं
Answer
Correct option: B.
$\frac{\pi}{4}$
B
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MCQ 3771 Mark
$\int_0^{\frac{\pi}{2}} \frac{d x}{1+\tan ^3 x}=$
  • A
    $0$
  • B
    1
  • C
    $\frac{\pi}{2}$
  • ✓
    $\frac{\pi}{4}$
Answer
Correct option: D.
$\frac{\pi}{4}$
D
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MCQ 3781 Mark
$\int_0^{\frac{\pi}{2}} \frac{\cot x}{1+\cot x} d x=$
  • A
    $\frac{\pi}{3}$
  • B
    $\frac{\pi}{2}$
  • ✓
    $\frac{\pi}{4}$
  • D
    $\frac{\pi}{6}$
Answer
Correct option: C.
$\frac{\pi}{4}$
C
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MCQ 3791 Mark
$\int_0^{\frac{\pi}{2}} \frac{1}{1+\sqrt{\cot x}} d x=$
  • A
    $0$
  • B
    $\frac{\pi}{2}$
  • ✓
    $\frac{\pi}{4}$
  • D
    $\pi$
Answer
Correct option: C.
$\frac{\pi}{4}$
C
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MCQ 3801 Mark
$\int_0^{\frac{\pi}{2}} \frac{1}{1+\cot x} d x=$
  • A
    $\frac{\pi}{2}$
  • B
    $0$
  • ✓
    $\frac{\pi}{4}$
  • D
    $\frac{\pi}{3}$
Answer
Correct option: C.
$\frac{\pi}{4}$
C
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MCQ 3821 Mark
$\int_0^{\frac{\pi}{2}} \cos ^5 x \sin x d x=$
  • A
    6
  • ✓
    $\frac{1}{6}$
  • C
    2
  • D
    3
Answer
Correct option: B.
$\frac{1}{6}$
B
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MCQ 3831 Mark
$\int_0^a \frac{f(x)}{f(x)+f(a-x)} d x=$
  • A
    a
  • B
    $\frac{\pi}{4}$
  • C
    $2 a$
  • ✓
    $\frac{a}{2}$
Answer
Correct option: D.
$\frac{a}{2}$
D
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MCQ 3841 Mark
$\int_0^a f(a-x) d x=$
  • ✓
    $\int_0^a f(x) d x$
  • B
    $\int_0^a f(-x) d x$
  • C
    $\int_0^a f(a) d x$
  • D
    $0$
Answer
Correct option: A.
$\int_0^a f(x) d x$
A
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MCQ 3851 Mark
$\int_0^3 \frac{1}{\sqrt{9-x^2}} d x=$
  • A
    $\frac{\pi}{6}$
  • B
    $-\frac{\pi}{6}$
  • ✓
    $\frac{\pi}{2}$
  • D
    $-\frac{\pi}{2}$
Answer
Correct option: C.
$\frac{\pi}{2}$
C
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MCQ 3861 Mark
$\int_0^2 \sqrt{6 x+4} d x=?$
  • A
    $\frac{64}{9}$
  • B
    7
  • ✓
    $\frac{56}{9}$
  • D
    $\frac{60}{9}$
Answer
Correct option: C.
$\frac{56}{9}$
C
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MCQ 3871 Mark
$\int_0^{2 a} f(x) d x=$
  • A
    $0$
  • B
    $2 \int_0^a f(x) d x$
  • ✓
    $\int_0^a[f(x)+f(2 a-x)] d x$
  • D
    $\int_0^a[f(x)+f(a-x)] d x$
Answer
Correct option: C.
$\int_0^a[f(x)+f(2 a-x)] d x$
C
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MCQ 3881 Mark
$\int_0^1\left(1+e^x\right) d x=$
  • A
    $1+\frac{1}{e}$
  • B
    $1-\frac{1}{e}$
  • C
    $\frac{2 e-1}{e}$
  • ✓
    e
Answer
Correct option: D.
e
D
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MCQ 3891 Mark
$\frac{d}{d x} \int f(x) d x$ किसके समान है?
  • A
    $f^{\prime}(x)$
  • B
    $f^{\prime}(x)+C$
  • C
    $f^{\prime \prime}(x)$
  • ✓
    $f(x)+C$
Answer
Correct option: D.
$f(x)+C$
D
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