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MCQ

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10 questions · auto-graded multiple-choice test.

MCQ 11 Mark
$\int(1-x)^{-2} d x=$
  • A
    $(1+x)^{-1}+c$
  • $(1-x)^{-1}+c$
  • C
    $(1-x)^{-1}-1+c$
  • D
    $(1-x)^{-1}+1+c$
Answer
Correct option: B.
$(1-x)^{-1}+c$
(b) $(1-x)^{-1}+c$
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MCQ 21 Mark
$\int\left(\frac{e^{2 x}+e^{-2 x}}{e^x}\right) d x$
  • $e^x-\frac{1}{3 e^{3 x}}+c$
  • B
    $e^x+\frac{1}{3 e^{3 x}}+c$
  • C
    $e^{-x}+\frac{1}{3 e^{3 x}}+c$
  • D
    $e^{-x}-\frac{1}{3 e^{3 x}}+c$
Answer
Correct option: A.
$e^x-\frac{1}{3 e^{3 x}}+c$
(a) $e^x-\frac{1}{3 e^{3 x}}+c$
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MCQ 31 Mark
$\int\left(x+\frac{1}{x}\right)^3 d x=$
  • A
    $\frac{1}{4}\left(x+\frac{1}{x}\right)^4+c$
  • $\frac{x^4}{4}+\frac{3 x^2}{2}+3 \log x-\frac{1}{2 x^2}+c$
  • C
    $\frac{x^4}{4}+\frac{3 x^2}{2}+3 \log x+\frac{1}{x^2}+c$
  • D
    $\left(x-x^{-1}\right)^3+c$
Answer
Correct option: B.
$\frac{x^4}{4}+\frac{3 x^2}{2}+3 \log x-\frac{1}{2 x^2}+c$
(b) $\frac{x^4}{4}+\frac{3 x^2}{2}+3 \log x-\frac{1}{2 x^2}+c$
Hint:
$
\left(x+\frac{1}{x}\right)^3=x^3+3 x+\frac{3}{x}+\frac{1}{x^3}
$
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MCQ 41 Mark
$\int \frac{d x}{(x-8)(x+7)}=$
  • A
    $\frac{1}{15} \log \left|\frac{x+2}{x-1}\right|+c$
  • B
    $\frac{1}{15} \log \left|\frac{x+8}{x+7}\right|+c$
  • $\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c$
  • D
    $(x-8)(x-7)+c$
Answer
Correct option: C.
$\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c$
(C) $\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c$
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MCQ 51 Mark
$\int \frac{d x}{\left(x-x^2\right)}=$
  • $\log x-\log (1-x)+c$
  • B
    $\log \left(1-x^2\right)+c$
  • C
    $-\log x+\log (1-x)+c$
  • D
    $\log \left(x-x^2\right)+c$
Answer
Correct option: A.
$\log x-\log (1-x)+c$
(a) $\log x-\log (1-x)+c$
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MCQ 61 Mark
$\int \frac{x+2}{2 x^2+6 x+5} d x=p \int \frac{4 x+6}{2 x^2+6 x+5} d x+\frac{1}{2} \int \frac{d x}{2 x^2+6 x+5}$, then $p =$
  • A
    $\frac{1}{3}$
  • B
    $\frac{1}{2}$
  • $\frac{1}{4}$
  • D
    $2$
Answer
Correct option: C.
$\frac{1}{4}$
(C) $\frac{1}{4}$
Hint:
$
\int \frac{x+2}{2 x^2+6 x+5} d x=\int \frac{\frac{1}{4}(4 x+6)+\frac{1}{2}}{2 x^2+6 x+5} d x
$
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MCQ 71 Mark
$\int x^2(3)^{x^3} d x=$
  • A
    $(3)^{x^3}+c$
  • $\frac{(3)^{x^3}}{3 \cdot \log 3}+c$
  • C
    $\log 3(3)^{x^3}+c$
  • D
    $x^2(3)^{x 3}$
Answer
Correct option: B.
$\frac{(3)^{x^3}}{3 \cdot \log 3}+c$
(b) $\frac{(3)^{x^3}}{3 \cdot \log 3}+c$
Hint:
Put $x^3=t$
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MCQ 81 Mark
$\int \sqrt{1+x^2} d x=$
  • $\frac{x}{2} \sqrt{1+x^2}+\frac{1}{2} \log \left(x+\sqrt{1+x^2}\right)+c$
  • B
    $\frac{2}{3}\left(1+x^2\right)^{3 / 2}+c$
  • C
    $\frac{1}{3}\left(1+x^2\right)+c$
  • D
    $\frac{(x)}{\sqrt{1+x^2}}+c$
Answer
Correct option: A.
$\frac{x}{2} \sqrt{1+x^2}+\frac{1}{2} \log \left(x+\sqrt{1+x^2}\right)+c$
(a) $\frac{x}{2} \sqrt{1+x^2}+\frac{1}{2} \log \left(x+\sqrt{1+x^2}\right)+c$
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MCQ 91 Mark
$\int \frac{\left(x^3+3 x^2+3 x+1\right)}{(x+1)^5} d x=$
  • $\frac{-1}{x+1}+c$
  • B
    $\left(\frac{-1}{x+1}\right)^5+c$
  • C
    $\log (x+1)+c$
  • D
    $\log |x+1|^5+c$
Answer
Correct option: A.
$\frac{-1}{x+1}+c$
(a) $\frac{-1}{x+1}+c$
Hint:
$
x^3+3 x^2+3 x+1=(x+1)^3
$
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MCQ 101 Mark
The value of $\int \frac{d x}{\sqrt{1-x}}$ is
  • A
    $20 \sqrt{1-x}+ C$
  • $-2 \sqrt{1-x}+c$
  • C
    $\sqrt{x}+ C$
  • D
    $x + C$
Answer
Correct option: B.
$-2 \sqrt{1-x}+c$
(b) $-2 \sqrt{1-x}+ c$
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