CBSE Class 12 Mathematics Integrals MCQs Set 04

Question. Evaluate : \( \int \frac{\sec^2 x}{2 + \tan x} \, dx \)
(a) \( \log|\tan x| + C \)
(b) \( \log|2 - \tan x| + C \)
(c) \( \log|2 + \tan x| + C \)
(d) None of the options
Answer: (c) \( \log|2 + \tan x| + C \)

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

Question. Evaluate : \( \int \frac{(a^x + b^x)^2}{a^x b^x} \, dx \)
(a) \( \frac{(a/b)^x}{\log(a/b)} + \frac{(b/a)^x}{\log(b/a)} + 2x + C, a \neq b \)
(b) \( \frac{(a/b)^x}{\log(a/b)} - \frac{(b/a)^x}{\log(b/a)} + 2x + C, a \neq b \)
(c) \( \left(\frac{a}{b}\right)^x + \left(\frac{b}{a}\right)^x + 2x + C, a \neq b \)
(d) None of the options
Answer: (a) \( \frac{(a/b)^x}{\log(a/b)} + \frac{(b/a)^x}{\log(b/a)} + 2x + C, a \neq b \)

Question. Find the value of \( \int_{-\pi/2}^{\pi/2} |\sin x| \, dx \).
(a) \( 0 \)
(b) \( 1 \)
(c) \( 2 \)
(d) \( 3 \)
Answer: (c) \( 2 \)

Question. Evaluate : \( \int_{0}^{1} \frac{x\tan^{-1} x}{(1+x^2)^{3/2}} \, dx \)
(a) \( \frac{4-\pi}{2\sqrt{2}} \)
(b) \( \frac{4+\pi}{2\sqrt{2}} \)
(c) \( \frac{4-\pi}{4\sqrt{2}} \)
(d) None of the options
Answer: (c) \( \frac{4-\pi}{4\sqrt{2}} \)

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

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

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

Question. Evaluate : \( \int \left( 5x^3 + 2x^{-5} - 7x + \frac{1}{\sqrt{x}} + \frac{5}{x} \right) \, dx \)
(a) \( \frac{5x^4}{4} - \frac{1}{2x^4} - \frac{7x^2}{2} + 2\sqrt{x} - 5\log|x| + C \)
(b) \( \frac{5x^4}{4} - \frac{1}{2x^4} - \frac{7x^2}{2} + 2\sqrt{x} + 5\log|x| + C \)
(c) \( \frac{5x^4}{4} + \frac{1}{2x^4} + \frac{7x^2}{2} + 2\sqrt{x} + 5\log|x| + C \)
(d) \( \frac{5x^4}{4} + \frac{1}{2x^4} + \frac{7x^2}{2} + 2\sqrt{x} - 5\log|x| + C \)
Answer: (b) \( \frac{5x^4}{4} - \frac{1}{2x^4} - \frac{7x^2}{2} + 2\sqrt{x} + 5\log|x| + C \)

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

Question. Evaluate : \( \int \frac{dx}{5 - 8x - x^2} \)
(a) \( \frac{1}{\sqrt{21}}\log\left| \frac{\sqrt{21}+x+4}{\sqrt{21}-x-4} \right| + C \)
(b) \( \frac{1}{2\sqrt{21}}\log\left| \frac{\sqrt{21}+x+4}{\sqrt{21}-x-4} \right| + C \)
(c) \( \frac{1}{\sqrt{21}}\log\left| \frac{\sqrt{21}-x-4}{\sqrt{21}+x+4} \right| + C \)
(d) \( \frac{1}{2\sqrt{21}}\log\left| \frac{\sqrt{21}-x-4}{\sqrt{21}+x+4} \right| + C \)
Answer: (b) \( \frac{1}{2\sqrt{21}}\log\left| \frac{\sqrt{21}+x+4}{\sqrt{21}-x-4} \right| + C \)

Question. Evaluate: \( \int [\sin(\log x) + \cos(\log x)] \, dx \)
(a) \( x\sin(\log x) + C \)
(b) \( \sin(\log x) + C \)
(c) \( x\cos(\log x) + C \)
(d) \( \cos(\log x) + C \)
Answer: (a) \( x\sin(\log x) + C \)

Question. Evaluate : \( \int \sec^2(7 - 4x) \, dx \)
(a) \( \frac{1}{4}\tan(7 - 4x) + C \)
(b) \( \frac{1}{4}\tan(7 + 4x) + C \)
(c) \( -\frac{1}{4}\tan(7 + 4x) + C \)
(d) \( -\frac{1}{4}\tan(7 - 4x) + C \)
Answer: (d) \( -\frac{1}{4}\tan(7 - 4x) + C \)

Question. Evaluate: \( \int \frac{x^3}{x+2} \, dx \)
(a) \( \frac{x^3}{3} - x^2 - 4x - 8\log|x+2| + C \)
(b) \( \frac{x^3}{3} - x^2 + 4x - 8\log|x+2| + C \)
(c) \( \frac{x^3}{3} + x^2 + 4x + 8\log|x+2| + C \)
(d) \( \frac{x^3}{3} + x^2 + 4x - 8\log|x+2| + C \)
Answer: (b) \( \frac{x^3}{3} - x^2 + 4x - 8\log|x+2| + C \)

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) \( \frac{a^x}{\log a} + \frac{x^{a+1}}{a+1} + \frac{a^a}{x} + C \)
(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) \( \frac{(2/5)^x}{\log_e(2/5)} + \frac{(3/5)^x}{\log_e(3/5)} + C \)
(b) \( \frac{(2/5)^x}{\log_e(2/5)} - \frac{(3/5)^x}{\log_e(3/5)} + C \)
(c) \( \frac{(2/5)^x}{\log_e(3/5)} + \frac{(3/5)^x}{\log_e(2/5)} + C \)
(d) None of the options
Answer: (a) \( \frac{(2/5)^x}{\log_e(2/5)} + \frac{(3/5)^x}{\log_e(3/5)} + C \)

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

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

Case Based MCQs

Case I : Read the following passage and answer the questions.
When the integrand can be expressed as a product of two functions, one of which can be differentiated and the other can be integrated, then we apply integration by parts.
If \( f(x) \) = first function (that can be differentiated) and \( g(x) \) = second function (that can be integrated), then the preference of this order can be decided by the word "ILATE", where
I stands for Inverse Trigonometric Function
L stands for Logarithmic Function
A stands for Algebraic Function
T stands for Trigonometric Function
E stands for Exponential Function, then
\( \int f(x)g(x)dx = f(x)\int g(x)dx - \int \left\{ \frac{d}{dx}f(x) \int g(x)dx \right\}dx \)

Question. \( \int x \sin 3x dx = \)
(a) \( -\frac{x \cos 3x}{3} + \frac{\sin 3x}{3} + c \)
(b) \( -\frac{x \cos 3x}{3} + \frac{\sin 3x}{9} + c \)
(c) \( \frac{x \cos 3x}{3} + \frac{\sin 3x}{9} + c \)
(d) \( -\frac{x \cos 3x}{3} - \frac{\sin 3x}{9} + c \)
Answer: (b) \( -\frac{x \cos 3x}{3} + \frac{\sin 3x}{9} + c \)

Question. \( \int \log(x+1) dx = \)
(a) \( \log(x+1) - x + c \)
(b) \( x \log(x+1) - x + c \)
(c) \( x \log(x+1) - \log(x+1) + x + c \)
(d) \( x \log(x+1) + \log(x+1) - x + c \)
Answer: (d) \( x \log(x+1) + \log(x+1) - x + c \)

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

Question. \( \int x^2 e^{3x} dx = \)
(a) \( \frac{e^{3x}}{9} (9x^2 + 6x + 2) + c \)
(b) \( \frac{e^{3x}}{9} (9x^2 - 6x + 2) + c \)
(c) \( \frac{e^{3x}}{27} (9x^2 + 6x + 2) + c \)
(d) \( \frac{e^{3x}}{27} (9x^2 - 6x + 2) + c \)
Answer: (d) \( \frac{e^{3x}}{27} (9x^2 - 6x + 2) + c \)

Question. \( \int (f(x)g''(x) - f''(x)g(x)) dx = \)
(a) \( f(x)g'(x) - f'(x)g(x) + c \)
(b) \( f(x)g'(x) + f'(x)g(x) + c \)
(c) \( f'(x)g(x) - f(x)g'(x) + c \)
(d) \( \frac{f(x)}{g'(x)} + c \)
Answer: (a) \( f(x)g'(x) - f'(x)g(x) + c \)

Case II : Read the following passage and answer the questions from 51 to 55.
Let \( f \) be a continuous function defined on the closed interval \( [a, b] \) and \( F \) be an antiderivative of \( f \), then \( \int_a^b f(x)dx = [F(x)]_a^b = F(b) - F(a) \)
This result is very useful as it gives us a method of calculating the definite integral easily. Here, we have no need to write integration constant \( c \) because if, we will write \( F(x) + c \), instead of \( F(x) \), we get
\( \int_a^b f(x)dx = [F(x) + c]_a^b = F(b) + c - F(a) - c = F(b) - F(a) \)

Question. Evaluate: \( \int_{\pi/4}^{\pi/2} \cos 2x dx \)
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{2} \)
(c) \( -\frac{1}{4} \)
(d) \( -\frac{1}{2} \)
Answer: (d) \( -\frac{1}{2} \)

Question. Evaluate : \( \int_1^2 \frac{dx}{x^2} \)
(a) \( \frac{1}{2} \)
(b) \( 1 \)
(c) \( 2 \)
(d) \( -1 \)
Answer: (a) \( \frac{1}{2} \)

Question. \( \int_{-1}^0 \frac{dx}{2x+3} \) is equal to
(a) \( \log \frac{3}{2} \)
(b) \( \log 3 - \log 1 \)
(c) \( \frac{log 3}{2} \)
(d) \( \log 3 + \log 1 \)
Answer: (c) \( \frac{log 3}{2} \)

Question. \( \int_1^3 (x-1)(x-2)(x-3) dx \) is equal to
(a) \( 3 \)
(b) \( 2 \)
(c) \( 1 \)
(d) \( 0 \)
Answer: (d) \( 0 \)

Question. \( \int_4^5 e^x dx \) equals
(a) \( e^5 - e^4 \)
(b) \( e^4 - e^5 \)
(c) \( e^9 \)
(d) \( e^{20} \)
Answer: (a) \( e^5 - e^4 \)

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: \( I = \int_0^1 \frac{dx}{\sqrt[3]{1 + x^3}} = \int_0^{2^{-1/3}} \frac{dt}{1 - t^3} \)
Reason: The integrand of the integral \( I \) becomes rational by the substitution \( t = \frac{x}{\sqrt[3]{1+x^3}} \)
(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. Assertion: \( \int_0^{2\pi} \sin^3 x dx = 0 \)
Reason: \( \sin^3 x \) is an odd function.
(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: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Assertion: The value of \( \int_0^{\pi/2} \sin^6 x dx = \frac{5\pi}{16} \)
Reason: If \( n \) is even, then \( \int_0^{\pi/2} \sin^n x dx \) equals \( \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \frac{n-5}{n-4} \dots \frac{1}{2} \cdot \frac{\pi}{2} \)
(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.