Sample QuestionsIntegrals questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
$\int_0^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x$ is equal to :
View full solution →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 $\int_1^e \log x d x$ is
View full solution →The value of $\int_0^3 \frac{d x}{\sqrt{9-x^2}}$ is:
- A
$\frac{\pi}{6}$
- B
$\frac{\pi}{4}$
- C
$\frac{\pi}{2}$
- D
$\frac{\pi}{18}$
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?
Answer: A.
View full solution →Assertion $(A): \int_2^8 \frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}} d x=3$
Reason (R): $\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$
- A
Both assertion (A) and reason (R) are true and reason $(R)$ is the correct explanation of assertion $(A)$.
- B
Both assertion (A) and reason (R) are true, but reason $(R)$ is not the correct explanation of the assertion (A).
- C
Assertion (A) is true and reason $(R)$ is false.
- D
Assertion $(A)$ is false, but reason $(R)$ is true.
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.
- ✓
Both (A) and (R) are true and (R) is the correct explanation of (A).
- B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
Answer: A.
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 $t=\frac{x}{\sqrt[3]{1+x^3}}$.
- A
Both (A) and (R) are true and (R) is the correct explanation of (A).
- B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
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 $x$.
- A
Both (A) and (R) are true and (R) is the correct explanation of (A).
- B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
View full solution →Assertion (A) :$\int \sin 3 x \cos 5 x d x=\frac{-\cos 8 x}{16}+\frac{\cos 2 x}{4}+C$
Reason (R) :$2 \cos A \sin B=\sin (A+B)-\sin (A-B)$
- ✓
Both (A) and (R) are true and (R) is the correct explanation of (A).
- B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
Answer: A.
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 = t6]
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 =t2) 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 →