CBSE Class 12 Mathematics Integrals MCQs Set 03

Question. Evaluate : \( \int (3\sin x - 2\cos x + 4\sec^2 x - 5\text{cosec}^2 x) \, dx \)
(a) \( -3\cos x - 2\sin x + 4\tan x + 5\cot x + C \)
(b) \( 3\cos x + 2\sin x + 4\tan x + 5\cot x + C \)
(c) \( -3\cos x + 2\sin x - 4\tan x - 5\cot x + C \)
(d) \( -3\cos x - 2\sin x - 4\tan x - 5\cot x + C \)
Answer: (a) \( -3\cos x - 2\sin x + 4\tan x + 5\cot x + C \)

Question. Evaluate : \( \int (2^x + 2^{-x})^2 \, dx \)
(a) \( \frac{1}{2\log 2}(2^{2x} - 2^{-2x}) + C \)
(b) \( \frac{1}{2\log 2}(2^{2x} - 2^{-2x}) + 2x + C \)
(c) \( \frac{1}{2\log 2}(2^{2x} + 2^{-2x}) + 2x + C \)
(d) \( \frac{1}{2\log 2}(2^{2x} + 2^{-2x}) + C \)
Answer: (b) \( \frac{1}{2\log 2}(2^{2x} - 2^{-2x}) + 2x + C \)

Question. Find the value of \( \int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} \, dx \).
(a) \( \tan x - \cot x + C \)
(b) \( -\tan x + \cot x + C \)
(c) \( \tan x + \cot x + C \)
(d) \( -\tan x - \cot x + C \)
Answer: (c) \( \tan x + \cot x + C \)

Question. Evaluate : \( \int_{2}^{4} \frac{x}{x^2 + 1} \, dx \)
(a) \( \frac{1}{2}\log\left(\frac{17}{5}\right) \)
(b) \( \frac{1}{2}\log\left(\frac{5}{17}\right) \)
(c) \( \log\left(\frac{17}{5}\right) \)
(d) \( \log\left(\frac{5}{17}\right) \)
Answer: (a) \( \frac{1}{2}\log\left(\frac{17}{5}\right) \)

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

Question. Evaluate : \( \int \frac{\cos x}{\left(\cos\frac{x}{2} + \sin\frac{x}{2}\right)^3} \, dx \)
(a) \( \frac{2}{\cos\frac{x}{2} + \sin\frac{x}{2}} + C \)
(b) \( \frac{-2}{\cos\frac{x}{2} - \sin\frac{x}{2}} + C \)
(c) \( \frac{-2}{\cos\frac{x}{2} + \sin\frac{x}{2}} + C \)
(d) \( \frac{2}{\cos\frac{x}{2} - \sin\frac{x}{2}} + C \)
Answer: (c) \( \frac{-2}{\cos\frac{x}{2} + \sin\frac{x}{2}} + C \)

Question. Evaluate: \( \int_{2}^{4} \frac{(x^2 + x)}{\sqrt{2x + 1}} \, dx \)
(a) \( 57 - 5\sqrt{5} \)
(b) \( \frac{57 - \sqrt{5}}{5} \)
(c) \( \frac{57 + 5\sqrt{5}}{5} \)
(d) \( \frac{57 - 5\sqrt{5}}{5} \)
Answer: (d) \( \frac{57 - 5\sqrt{5}}{5} \)

Question. Evaluate : \( \int 2^{2^{2^x}} 2^{2^x} 2^x \, dx \)
(a) \( \frac{1}{(\log 2)^3} 2^{2^{2^x}} + C \)
(b) \( \frac{1}{(\log 2)^3} 2^{2^x} + C \)
(c) \( \frac{1}{(\log 2)^2} 2^{2^x} + C \)
(d) \( \frac{1}{(\log 2)^4} 2^{2^{2^x}} + C \)
Answer: (a) \( \frac{1}{(\log 2)^3} 2^{2^{2^x}} + C \)

Question. Evaluate : \( \int 2^{(x+3)} \, dx \)
(a) \( \frac{2^x}{\log 2} + C \)
(b) \( \frac{2^3}{\log 2} + C \)
(c) \( \frac{2^{(x+3)}}{\log 2} + C \)
(d) \( \frac{2^{(x-3)}}{\log 2} + C \)
Answer: (c) \( \frac{2^{(x+3)}}{\log 2} + C \)

Question. Evaluate : \( \int \sin^3 x \cos^3 x \, dx \)
(a) \( -\frac{1}{32}\left\{ -3\cos 2x + \frac{1}{6}\cos 6x \right\} + C \)
(b) \( \frac{1}{32}\left\{ \cos 6x + \frac{1}{6}\cos 2x \right\} + C \)
(c) \( \frac{1}{32}\left\{ -3\cos 2x + \frac{1}{6}\cos 6x \right\} + C \)
(d) None of the options
Answer: (c) \( \frac{1}{32}\left\{ -3\cos 2x + \frac{1}{6}\cos 6x \right\} + C \)

Question. Evaluate : \( \int \sqrt{(x-3)(5-x)} \, dx \)
(a) \( \frac{1}{2}(x-4)\sqrt{(x-3)(5-x)} + \frac{1}{2}\cos^{-1}(x-4) + C \)
(b) \( \frac{1}{2}(x-4)\sqrt{(x-3)(5-x)} + \frac{1}{2}\sin^{-1}(x-4) + C \)
(c) \( \frac{1}{2}\sqrt{(x-3)(5-x)} + \frac{1}{2}\sin^{-1}(x-4) + C \)
(d) None of the options
Answer: (b) \( \frac{1}{2}(x-4)\sqrt{(x-3)(5-x)} + \frac{1}{2}\sin^{-1}(x-4) + C \)

Question. Evaluate : \( \int_{0}^{\pi/4} \tan^3 x \, dx \)
(a) \( (1 - \log 2) \)
(b) \( (1 + \log 2) \)
(c) \( \frac{1}{2}(1 - \log 2) \)
(d) \( \frac{1}{2}(1 + \log 2) \)
Answer: (c) \( \frac{1}{2}(1 - \log 2) \)

Question. Evaluate : \( \int \frac{\cot x}{\sqrt[3]{\sin x}} \, dx \)
(a) \( \frac{-3}{\sqrt[3]{\sin x}} + C \)
(b) \( \frac{-2}{\sin^3 x} + C \)
(c) \( \frac{3}{\sin^{1/3} x} + C \)
(d) None of the options
Answer: (a) \( \frac{-3}{\sqrt[3]{\sin x}} + C \)

Question. Evaluate : \( \int x^2(ax + b)^{-2} \, dx \)
(a) \( \frac{1}{a^3} \left[ ax + b - \frac{b^2}{ax + b} - 2b\log(ax + b) \right] + C \)
(b) \( \frac{1}{a^3} \left[ ax + b + \frac{b^2}{ax + b} - 2b\log(ax + b) \right] + C \)
(c) \( \frac{1}{a^3} \left[ ax + b + \frac{b^2}{ax + b} + 2b\log(ax + b) \right] + C \)
(d) \( \frac{1}{a^3} \left[ ax + b - \frac{b^2}{ax + b} + 2b\log(ax + b) \right] + C \)
Answer: (a) \( \frac{1}{a^3} \left[ ax + b - \frac{b^2}{ax + b} - 2b\log(ax + b) \right] + C \)

Question. Evaluate : \( \int_{0}^{1} \left\{ e^x + \sin\frac{\pi x}{4} \right\} \, dx \)
(a) \( e + 1 + \frac{2\sqrt{2}}{\pi} - \frac{4}{\pi} \)
(b) \( e - 1 - \frac{2\sqrt{2}}{\pi} + \frac{4}{\pi} \)
(c) \( e + 1 - \frac{2\sqrt{2}}{\pi} + \frac{4}{\pi} \)
(d) \( e - 1 + \frac{2\sqrt{2}}{\pi} - \frac{4}{\pi} \)
Answer: (b) \( e - 1 - \frac{2\sqrt{2}}{\pi} + \frac{4}{\pi} \)

Question. Evaluate : \( \int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx \)
(a) \( \frac{3}{2}\sin^{-1}\left(\frac{x}{a}\right)^{3/2} + C \)
(b) \( \frac{2}{3}\sin^{-1}\left(\frac{x}{a}\right)^{3/2} + C \)
(c) \( \frac{2}{3}\cos^{-1}\left(\frac{x}{a}\right)^{3/2} + C \)
(d) \( \frac{3}{2}\cos^{-1}\left(\frac{x}{a}\right)^{3/2} + C \)
Answer: (b) \( \frac{2}{3}\sin^{-1}\left(\frac{x}{a}\right)^{3/2} + C \)

Question. Evaluate : \( \int_{0}^{2} e^{3 - 4x} \, dx \)
(a) \( -\frac{1}{4}(e^5 - e^3) \)
(b) \( \frac{1}{4}(e^5 - e^3) \)
(c) \( \frac{1}{4}(e^{-5} - e^3) \)
(d) \( -\frac{1}{4}(e^{-5} - e^3) \)
Answer: (d) \( -\frac{1}{4}(e^{-5} - e^3) \)

Question. The value of \( \int_{0}^{2\pi} \frac{dx}{e^{\sin x} + 1} \) is
(a) \( \pi \)
(b) \( 0 \)
(c) \( 3\pi \)
(d) \( \frac{\pi}{2} \)
Answer: (a) \( \pi \)

Question. Evaluate : \( \int \frac{10x^9 + 10^x \log_e 10}{10^x + x^{10}} \, dx \)
(a) \( \log_e(10^x - x^{10}) + C \)
(b) \( \log_e(10^x + x^{10}) + C \)
(c) \( \log_e(10^x + x^9) + C \)
(d) \( \log_e(10^x - x^9) + C \)
Answer: (b) \( \log_e(10^x + x^{10}) + C \)

Question. Evaluate : \( \int \frac{1}{\sin x + \sqrt{3}\cos x} \, dx \)
(a) \( \frac{1}{2}\log\left| \tan\left(\frac{x}{2} + \frac{\pi}{6}\right) \right| + C \)
(b) \( \frac{1}{2}\log\left| \tan\frac{x}{2} \right| + C \)
(c) \( \frac{1}{2}\log\left| \tan\left(\frac{x}{2} - \frac{\pi}{6}\right) \right| + C \)
(d) \( \frac{1}{2}\log\left| \tan\left(x - \frac{\pi}{6}\right) \right| + C \)
Answer: (a) \( \frac{1}{2}\log\left| \tan\left(\frac{x}{2} + \frac{\pi}{6}\right) \right| + C \)

Case Based MCQs

Case I : Read the following passage and answer the questions.
Integration is the process of finding the anti-derivative of a function. In this process, we are provided with the derivative of a function and asked to find out the function (i.e., Primitive). Integration is the inverse process of differentiation.
Let \( f(x) \) be a function of \( x \). If there is a function \( g(x) \), such that \( \frac{d}{dx}(g(x)) = f(x) \), then \( g(x) \) is called an integral of \( f(x) \) w.r.t. \( x \) and is denoted by \( \int f(x) \, dx = g(x) + c \), where \( c \) is constant of integration.

Question. \( \int (3x + 4)^3 \, dx \) is equal to
(a) \( \frac{(3x+4)^4}{12} + c \)
(b) \( \frac{3(3x+4)^4}{4} + c \)
(c) \( \frac{3(3x+4)^2}{2} + c \)
(d) \( \frac{3(3x+4)^2}{4} + c \)
Answer: (a) \( \frac{(3x+4)^4}{12} + c \)

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

Question. \( \int \sin^2 x \, dx \) is equal to
(a) \( \frac{x}{2} + \frac{\sin 2x}{4} + c \)
(b) \( \frac{x}{2} - \frac{\sin 2x}{4} + c \)
(c) \( x + \frac{\sin 2x}{2} + c \)
(d) \( x - \frac{\sin 2x}{2} + c \)
Answer: (b) \( \frac{x}{2} - \frac{\sin 2x}{4} + c \)

Question. Evaluate : \( \int \frac{\sin x}{1 + \sin x} dx \)
(a) \( \sec x - \tan x + C \)
(b) \( \sec x + \tan x + x + C \)
(c) \( \sec x + \tan x + C \)
(d) \( \sec x - \tan x + x + C \)
Answer: (d) \( \sec x - \tan x + x + C \)

Question. Evaluate: \( \int_{-\pi}^{\pi} x^{10} \sin^7 x dx \)
(a) \( 1 \)
(b) \( 2 \)
(c) \( -1 \)
(d) \( 0 \)
Answer: (d) \( 0 \)

Question. Evaluate: \( \int (e^{x \log a} + e^{a \log x} + e^{a \log a}) dx \)
(a) \( a^x \log a + \frac{x^{a+1}}{a+1} + \frac{a^a}{x} + c \)
(b) \( a^x \log a + (a+1) x^{a+1} + a^a x + c \)
(c) \( \frac{a^x}{\log a} + \frac{x^{a+1}}{a+1} + a^a x + c \)
(d) None of the options
Answer: (c) \( \frac{a^x}{\log a} + \frac{x^{a+1}}{a+1} + a^a x + c \)

Question. Evaluate : \( \int \frac{2^x + 3^x}{5^x} dx \)
(a) \( \frac{\left(\frac{2}{5}\right)^x}{\log_e\left(\frac{2}{5}\right)} + \frac{\left(\frac{3}{5}\right)^x}{\log_e\left(\frac{3}{5}\right)} + c \)
(b) \( \frac{\left(\frac{2}{5}\right)^x}{\log_e\left(\frac{2}{5}\right)} - \frac{\left(\frac{3}{5}\right)^x}{\log_e\left(\frac{3}{5}\right)} + c \)
(c) \( \frac{\left(\frac{2}{5}\right)^x}{\log_e\left(\frac{5}{2}\right)} + \frac{\left(\frac{3}{5}\right)^x}{\log_e\left(\frac{5}{3}\right)} + c \)
(d) None of the options
Answer: (a) \( \frac{\left(\frac{2}{5}\right)^x}{\log_e\left(\frac{2}{5}\right)} + \frac{\left(\frac{3}{5}\right)^x}{\log_e\left(\frac{3}{5}\right)} + c \)

Question. Evaluate : \( \int_0^2 (x - [x]) dx \)
(a) \( 0 \)
(b) \( -1 \)
(c) \( 1 \)
(d) \( 2 \)
Answer: (c) \( 1 \)

Question. Evaluate : \( \int \frac{(x^4 - x)^{\frac{1}{4}}}{x^5} dx \)
(a) \( \frac{4}{15} \left(1 - \frac{1}{x^3}\right)^{\frac{5}{4}} + C \)
(b) \( \frac{-4}{15} \left(1 - \frac{1}{x^3}\right)^{\frac{5}{4}} + C \)
(c) \( \frac{2}{15} \left(1 - \frac{1}{x^3}\right)^{\frac{5}{4}} + C \)
(d) \( \frac{-2}{15} \left(1 - \frac{1}{x^3}\right)^{\frac{5}{4}} + C \)
Answer: (a) \( \frac{4}{15} \left(1 - \frac{1}{x^3}\right)^{\frac{5}{4}} + C \)

Assertion & Reasoning Based MCQs

Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given. Choose the correct answer out of the following choices
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: \( \int \sin 3x \cos 5x dx = \frac{-\cos 8x}{16} + \frac{\cos 2x}{4} + C \)
Reason: \( 2\cos A \sin B = \sin(A+B) - \sin(A-B) \)
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. Let \( F(x) \) be an indefinite integral of \( \sin^2 x \).
Assertion: The function \( F(x) \) satisfies \( F(x + \pi) = F(x) \) for all real \( x \).
Reason: \( \sin^2(x + \pi) = \sin^2 x \) for all real \( x \).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (d) Assertion is wrong statement but Reason is correct statement.