CBSE Class 12 Mathematics Integrals MCQs Set 01

Question. \( \int e^x (\cos x - \sin x) dx \) is equal to
(a) \( e^x \cos x + C \)
(b) \( e^x \sin x + C \)
(c) \( -e^x \cos x + C \)
(d) \( -e^x \sin x + C \)
Answer: (a) \( e^x \cos x + C \)

Question. \( \int \frac{dx}{\sin^2 x \cos^2 x} \) is equal to
(a) \( \tan x + \cot x + C \)
(b) \( (\tan x + \cot x)^2 + C \)
(c) \( \tan x - \cot x + C \)
(d) \( (\tan x - \cot x)^2 + C \)
Answer: (c) \( \tan x - \cot x + C \)

Question. If \( \int \frac{3e^x - 5e^{-x}}{4e^x + 5e^{-x}} dx = ax + b \log |4e^x + 5e^{-x}| + C \) then
(a) \( a = \frac{-1}{8}, b = \frac{7}{8} \)
(b) \( a = \frac{1}{8}, b = \frac{7}{8} \)
(c) \( a = \frac{-1}{8}, b = \frac{-7}{8} \)
(d) \( a = \frac{1}{8}, b = \frac{-7}{8} \)
Answer: (c) \( a = \frac{-1}{8}, b = \frac{-7}{8} \)

Question. \( \int_{a+c}^{b+c} f(x) dx \) is equal to
(a) \( \int_{a}^{b} f(x - c) dx \)
(b) \( \int_{a}^{b} f(x + c) dx \)
(c) \( \int_{a}^{b} f(x) dx \)
(d) \( \int_{a-c}^{b-c} f(x) dx \)
Answer: (a) \( \int_{a}^{b} f(x + c) dx \)

Question. If \( f \) and \( g \) are continuous functions in \( [0, 1] \) satisfying \( f(x) = f(a - x) \) and \( g(x) + g(a - x) = a \), then \( \int_{0}^{a} f(x).g(x) dx \) is equal to
(a) \( \frac{a}{2} \)
(b) \( \frac{a}{2} \int_{0}^{a} f(x) dx \)
(c) \( \int_{0}^{a} f(x) dx \)
(d) \( a \int_{0}^{a} f(x) dx \)
Answer: (b) \( \frac{a}{2} \int_{0}^{a} f(x) dx \)

Question. If \( x = \int_{0}^{y} \frac{dt}{\sqrt{1 + 9t^2}} \) and \( \frac{d^2 y}{dx^2} = ay \), then \( a \) is equal to
(a) 3
(b) 6
(c) 9
(d) 1
Answer: (c) 9

Question. \( \int_{-1}^{1} \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1} dx \) is equal to
(a) \( \log 2 \)
(b) \( 2 \log 2 \)
(c) \( \frac{1}{2} \log 2 \)
(d) \( 4 \log 2 \)
Answer: (b) \( 2 \log 2 \)

Question. \( \int_{0}^{1} \frac{e^t}{1 + t} dt = a \), then \( \int_{0}^{1} \frac{e^t}{(1 + t)^2} dt \) is equal to
(a) \( a - 1 + \frac{e}{2} \)
(b) \( a + 1 - \frac{e}{2} \)
(c) \( a - 1 - \frac{e}{2} \)
(d) \( a + 1 + \frac{e}{2} \)
Answer: (b) \( a + 1 - \frac{e}{2} \)

Question. \( \int \frac{\cos 2x - \cos 2\theta}{\cos x - \cos \theta} dx \) is equal to
(a) \( 2(\sin x + x \cos \theta) + C \)
(b) \( 2(\sin x - x \cos \theta) + C \)
(c) \( 2(\sin x + 2x \cos \theta) + C \)
(d) \( 2(\sin x - 2x \cos \theta) + C \)
Answer: (c) \( 2(\sin x + x \cos \theta) + C \)

Question. \( \int \frac{dx}{\sin (x - a) \sin (x - b)} \) is equal to
(a) \( \sin(b - a) \log \left| \frac{\sin (x - b)}{\sin (x - a)} \right| + C \)
(b) \( \csc(b - a) \log \left| \frac{\sin (x - a)}{\sin (x - b)} \right| + C \)
(c) \( \csc(b - a) \log \left| \frac{\sin (x - b)}{\sin (x - a)} \right| + C \)
(d) \( \sin(b - a) \log \left| \frac{\sin (x - a)}{\sin (x - b)} \right| + C \)
Answer: (a) \( \csc(b - a) \log \left| \frac{\sin (x - b)}{\sin (x - a)} \right| + C \)

Question. \( \int \tan^{-1} \sqrt{x} dx \) is equal to
(a) \( (x + 1) \tan^{-1} \sqrt{x} - \sqrt{x} + C \)
(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: (a) \( (x + 1) \tan^{-1} \sqrt{x} - \sqrt{x} + C \)

Question. \( \int e^x \left( \frac{1 - x}{1 + x^2} \right)^2 dx \) is equal to
(a) \( \frac{e^x}{1 + x^2} + C \)
(b) \( \frac{-e^x}{1 + x^2} + C \)
(c) \( \frac{e^x}{(1 + x^2)^2} + C \)
(d) \( \frac{-e^x}{(1 + x^2)^2} + C \)
Answer: (c) \( \frac{e^x}{1 + x^2} + C \)

Question. \( \int \frac{x^9}{(4x^2 + 1)^6} dx \) is equal to
(a) \( \frac{1}{5x} \left( 4 + \frac{1}{x^2} \right)^{-5} + C \)
(b) \( \frac{1}{5} \left( 4 + \frac{1}{x^2} \right)^{-5} + C \)
(c) \( \frac{1}{10x} (1 + 4)^{-5} + C \)
(d) \( \frac{1}{10} \left( \frac{1}{x^2} + 4 \right)^{-5} + C \)
Answer: (d) \( \frac{1}{10} \left( \frac{1}{x^2} + 4 \right)^{-5} + C \)

Question. \( \int \frac{\sin^6 x}{\cos^8 x} dx \) is equal to
(a) \( \frac{\tan^6 x}{5} + C \)
(b) \( \frac{\tan^7 x}{5} + C \)
(c) \( \frac{\tan^7 x}{7} + C \)
(d) None of the options
Answer: (c) \( \frac{\tan^7 x}{7} + C \)

Question. \( \int_{-2}^{2} |x \cos \pi x| dx \) is equal to
(a) \( \frac{8}{\pi} \)
(b) \( \frac{4}{\pi} \)
(c) \( \frac{2}{\pi} \)
(d) \( \frac{1}{\pi} \)
Answer: (a) \( \frac{8}{\pi} \)

Question. The integral value of \( \int_{0}^{\pi/2} \frac{\tan x}{1 + m^2 \tan^2 x} dx \)
(a) \( \log \left( \frac{m}{m^2 - 1} \right) \)
(b) \( \log \left( \frac{m^2 - m}{2} \right) \)
(c) \( \log 3m \)
(d) 0
Answer: (a) \( \log \left( \frac{m}{m^2 - 1} \right) \)

Question. The integral of \( \int \frac{x}{\sqrt{x + 1}} dx \) is equal to
(a) \( 2 \left[ \frac{x\sqrt{x}}{3} - \frac{x}{2} + \sqrt{x} - \log |(\sqrt{x} + 1)| \right] + C \)
(b) \( \frac{x\sqrt{x}}{3} + \frac{x}{2} - \sqrt{x} + \log |(\sqrt{x} + 1)| + C \)
(c) \( \sqrt{x} - \log (\sqrt{x} + 1) + C \)
(d) None of the options
Answer: (a) \( 2 \left[ \frac{x\sqrt{x}}{3} - \frac{x}{2} + \sqrt{x} - \log |(\sqrt{x} + 1)| \right] + C \)

Question. If \( \int \frac{x^3}{\sqrt{1 + x^2}} dx = a(1 + x^2)^{3/2} + b\sqrt{1 + x^2} + C \), then
(a) \( a = \frac{1}{3}, b = -1 \)
(b) \( a = \frac{1}{3}, b = -1 \)
(c) \( a = \frac{-1}{3}, b = -1 \)
(d) \( a = \frac{1}{3}, b = -1 \)
Answer: (b) \( a = \frac{1}{3}, b = -1 \)

Question. If \( \int_{0}^{a} \frac{1}{1 + 4x^2} dx = \frac{\pi}{8} \) then find value of \( a \).
(a) \( \frac{-1}{2} \)
(b) \( \frac{7}{2} \)
(c) \( \frac{1}{2} \)
(d) 0
Answer: (c) \( \frac{1}{2} \)

Question. The integral \( \int \frac{x^9}{(4x^2 + 1)^6} dx \) is equal to
(a) \( \frac{1}{5x} \left( 4 + \frac{1}{x^2} \right)^{-5} + C \)
(b) \( \frac{1}{5} \left( 4 + \frac{1}{x^2} \right)^{-5} + C \)
(c) \( \frac{1}{10x} (5)^{-5} + C \)
(d) \( \frac{1}{10} \left( \frac{1}{x^2} + 4 \right)^{-5} + C \)
Answer: (d) \( \frac{1}{10} \left( \frac{1}{x^2} + 4 \right)^{-5} + C \)

Assertion-Reason Questions

Question. Assertion (A) : \( \int \frac{1}{1 + x^2} dx = \tan^{-1} x + C \), where \( C \) is an arbitrary constant.
Reason (R) : Since \( \frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2} \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.

Question. Assertion (A) : \( \int \frac{1}{2\sqrt{x}} dx = \sqrt{x} + C \)
Reason (R) : \( \int \cos x dx = \sin x + C \)
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.

Question. Assertion (A) : If \( f'(x) = x + \frac{1}{1 + x^2} \) and \( f(0) = 0 \) then \( f(x) = \frac{x^2}{2} + \tan^{-1} x \).
Reason (R) : \( \int x^n dx = \frac{x^{n+1}}{n + 1} + C \)
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.

Question. Assertion (A) : \( \int_{0}^{\pi/2} \cos x dx = 1 \)
Reason (R) : If \( f(x) \) is continuous in \( [a, b] \) and \( \int f(x) dx = \phi(x) \), then \( \int_{a}^{b} f(x) dx = \phi(b) - \phi(a) \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.

Question. Assertion (A) : \( \int_{0}^{1} \frac{1}{\sqrt{1 - x^2}} dx = \frac{\pi}{2} \)
Reason (R) : \( \int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \tan^{-1} \frac{x}{a} + C \)
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.