Sample QuestionsIntegrals questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $\int_{-2}^3 x^2 d x=k \int_0^2 x^2 d x+\int_2^3 x^2 d x$, then the value of $k$ is
View full solution →The value of $\int_0^3 \frac{d x}{\sqrt{9-x^2}}$ is:
View full solution →The value of $\int_1^e \log x d x$ is
View full solution →$\int_0^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x$ is equal to :
View full solution →Which of these is equal to $\int e^{(x \log 5)} e^x d x$, where $C$ is the constant of integration?
View full solution →Assertion $(A): \int_2^8 \frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}} d x=3$
Reason $(R): \int_a^b f(x) d x=\int_a^b f(a+b-x) d x$
View full solution →Assertion (A) : $\int_0^{2 \pi} \sin ^3 x d x=0$
Reason (R) : $\sin ^3 x$ is an odd function.
View full solution →Assertion (A) : The value of$\int_{-3}^3\left(a x^5+b x^3+c x+k\right) d x,$ where $a, b, c, k$ are constants, depends on only $k$.
Reason (R) : $\int_{-a}^a f(x) d x=0$, if $f(-x)=-f(x)$ i.e., $f$ is an odd function.
View full solution →Assertion $(A) : I=\int_0^1 \frac{d x}{\sqrt[3]{1+x^3}}=\int_0^{2^{-1 / 3}} \frac{d t}{1-t^3}$
Reason $(R) :$ The integrand of the integral $I$ becomes rational by the substitution $t=\frac{x}{\sqrt[3]{1+x^3}}$.
View full solution →Let $F(x)$ be an indefinite integral of $\sin ^2 x$.
Assertion (A) : The function $F(x)$ satisfies $F(x+\pi)=F(x)$ for all real $x$.
Reason (R) : $\sin ^2(x+\pi)=\sin ^2 x$ for all real $x$.
View full solution →Integrate the function: $\frac{\cos x}{\sqrt{4-\sin ^{2} x}}$
View full solution →Integrate the function $\int {\frac{{{e^{5\log x}} - {e^{4\log x}}}}{{{e^{3\log x}} - {e^{2\log x}}}}} dx$
View full solution →Integrate the function $\frac{\sin x}{\sin (x-a)}$
View full solution →Integrate the function $\frac{5 x}{(x+1)\left(x^{2}+9\right)}$
View full solution →Integrate the function $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$ [Hint: $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=\frac{1}{x^{\frac{1}{3}}\left(1+x^{\frac{1}{6}}\right)}$ Put $x = t^6]$
View full solution →Evaluate the integral $\int_{-1}^{1} \frac{d x}{x^{2}+2 x+5}$ using substitution.
View full solution →Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\cos ^{2} x} d x$ using substitution.
View full solution →Evaluate the integral $\int_{0}^{1} \frac{x}{x^{2}+1} d x$ using substitution.
View full solution →Evaluate the definite integral $\int_0^1 \frac{d x}{\sqrt{1-x^2}}$
View full solution →Evaluate the definite integral $\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{4}} {\cos ecxdx} $
View full solution →Evaluate the integral $\int\limits_1^2 {\left( {\frac{1}{x} - \frac{1}{{2{x^2}}}} \right){e^{2x}}dx} $ using substitution.
View full solution →Evaluate the integral $\int_{0}^{2} x \sqrt{x+2} ($Put $x + 2 =t^2)$ using substitution.
View full solution →Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos ^{5} \phi~ d \phi$ using substitution.
View full solution →Evaluate the definite integral $\int\limits_2^3 {\frac{{xdx}}{{{x^2} + 1}}} $
View full solution →Integrate the function $\sqrt {1 + \frac{{{x^2}}}{9}} $
View full solution →Evaluate the integral $\int_{0}^{2} \frac{d x}{x+4-x^{2}}$ using substitution.
View full solution →Evaluate the integral $\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$ using substitution.
View full solution →Evaluate the definite integral $\int _ { 0 } ^ { 1 } x e ^ { x ^ { 2 } } d x.$
View full solution →Integrate the function $\sqrt{1+3 x-x^{2}}$
View full solution →Integrate the rational function $\frac{3 x+5}{x^{3}-x^{2}-x+1}$
View full solution →Fill in the blanks:
$\int\frac{\text{x}+3}{(\text{x}+4)^2}\text{e}^\text{x}\text{dx}=$ ________.
View full solution →Fill in the blanks:
If $\int\limits^\text{a}_0\frac{1}{1+4\text{x}^2}\text{dx}=\frac{\pi}{8},$ then a = ________.
View full solution →Fill in the blanks:
$\int\frac{\sin\text{x}}{3+4\cos^2\text{x}}\text{dx}=$ ________.
View full solution →Fill in the blanks:
The value of $\int\limits^\pi_{-\pi}\sin^3\text{x}\cos^2\text{x dx}$ is _______.
View full solution →Fill in the blanks:
$\int\limits^{\frac{\pi}{2}}_0\cos\text{x e}^{\sin\text{x}}\text{dx}$ is equal to_________.
View full solution →Integrate the function: $\frac{\sqrt{\tan x}}{\sin x \cos x}$
View full solution →Integrate the function: $\frac{1}{{1 - \tan x}}$
View full solution →Integrate the function: $\frac{1}{1+\cot x}$
View full solution →By using the properties of definite integrals, evaluate the integral $\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$
View full solution →By using the properties of definite integrals, evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x d x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}$
View full solution →