JEE Mathematics Indefinite Integration MCQs Set 01

Choose the most appropriate option (a, b, c or d).

Question. If \( \phi(x) = \int \cot^4 x \, dx + \frac{1}{3} \cot^3 x - \cot x \) and \( \phi\left(\frac{\pi}{2}\right) = \frac{\pi}{2} \) then \( \phi(x) \) is
(a) \( \pi - x \)
(b) \( x - \pi \)
(c) \( \frac{\pi}{2} - x \)
(d) None of the options
Answer: (d) None of the options

Question. Integral of \( f(x) = \sqrt{1 + x^2} \) with respect to \( x^2 \) is
(a) \( \frac{2 (1 + x^2)^{3/2}}{3x} + k \)
(b) \( \frac{2}{3} (1 + x^2)^{3/2} + k \)
(c) \( \frac{2}{3} x(1 + x^2)^{3/2} + k \)
(d) None of the options
Answer: (b) \( \frac{2}{3} (1 + x^2)^{3/2} + k \)

Question. \( \int \frac{d(x^2 + 1)}{\sqrt{x^2 + 2}} \) is equal to
(a) \( 2\sqrt{x^2 + 2} + k \)
(b) \( \sqrt{x^2 + 2} + k \)
(c) \( \frac{1}{(x^2 + 2)^{3/2}} + k \)
(d) None of the options
Answer: (a) \( 2\sqrt{x^2 + 2} + k \)

Question. \( \int \cos\left\{ 2 \tan^{-1} \sqrt{\frac{1-x}{1+x}} \right\} dx \) is equal to
(a) \( \frac{1}{8}(x^2 - 1) + k \)
(b) \( \frac{1}{2}x^2 + k \)
(c) \( \frac{1}{2}x + k \)
(d) None of the options
Answer: (b) \( \frac{1}{2}x^2 + k \)

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

Question. Let the equation of a curve passing through the point (0, 1) be given by \( y = \int x^2 \cdot e^{x^3} dx \). If the equation of the curve is written in the form \( x = f(y) \) the \( f(y) \) is
(a) \( \sqrt{\log_e(3y - 2)} \)
(b) \( \sqrt[3]{\log_e(3y - 2)} \)
(c) \( \sqrt[3]{\log_e(2 - 3y)} \)
(d) None of the options
Answer: (b) \( \sqrt[3]{\log_e(3y - 2)} \)

Question. \( \int \frac{xdx}{1 + x^4} \) is equal to
(a) \( \tan^{-1} x^2 + k \)
(b) \( \frac{1}{2} \tan^{-1} x^2 + k \)
(c) \( \log (1 + x^4) + k \)
(d) None of the options
Answer: (b) \( \frac{1}{2} \tan^{-1} x^2 + k \)

Question. The antiderivative of \( \frac{2^x}{\sqrt{1 - 4^x}} \) w.r.t. \( x \) is
(a) \( \log_2 e \cdot \sin^{-1}(2^x) + k \)
(b) \( \sin^{-1}(2^x) + k \)
(c) \( \cos^{-1}(2^x) \cdot \frac{1}{\log_e 2} + k \)
(d) None of the options
Answer: (a) \( \log_2 e \cdot \sin^{-1}(2^x) + k \)

Question. \( \int \frac{(1 + x)^2}{x + x^3} dx \) is equal to
(a) \( \log_e x + \log_e(1 + x^2) + k \)
(b) \( \log_e x + \tan^{-1} x + k \)
(c) \( \log_e x + 2 \tan^{-1} x + k \)
(d) None of the options
Answer: (c) \( \log_e x + 2 \tan^{-1} x + k \)

Question. \( \int \frac{x^{5/2}}{\sqrt{1 + x^7}} dx \) is
(a) \( \frac{2}{7} \log(x^{7/2} + \sqrt{x^7 + 1}) + c \)
(b) \( \frac{1}{2} \log \frac{x^7 + 1}{x^7 - 1} + c \)
(c) \( 2\sqrt{1 + x^7} + c \)
(d) None of the options
Answer: (a) \( \frac{2}{7} \log(x^{7/2} + \sqrt{x^7 + 1}) + c \)

Question. \( \int \frac{dx}{x^{1/5}(1 + x^{4/5})^{1/2}} \) is
(a) \( \sqrt{1 + x^{4/5}} + k \)
(b) \( \frac{5}{2} \sqrt{1 + x^{4/5}} + k \)
(c) \( x^{4/5}(1 + x^{4/5})^{1/2} + k \)
(d) None of the options
Answer: (b) \( \frac{5}{2} \sqrt{1 + x^{4/5}} + k \)

Question. The primitive of the function \( x |\cos x| \) when \( \frac{\pi}{2} < x < \pi \) is given by
(a) \( \cos x + x \sin x \)
(b) \( -\cos x - x \sin x \)
(c) \( x \sin x - \cos x \)
(d) None of the options
Answer: (b) \( -\cos x - x \sin x \)

Question. \( \int x \sec x^2 \, dx \) is equal to
(a) \( \frac{1}{2} \log(\sec x^2 + \tan x^2) + k \)
(b) \( \frac{x^2}{2} \log(\sec x^2 + \tan x^2) + k \)
(c) \( 2 \log(\sec x^2 + \tan x^2) + k \)
(d) None of the options
Answer: (a) \( \frac{1}{2} \log(\sec x^2 + \tan x^2) + k \)

Question. \( \int \frac{f(x) \phi'(x) - f'(x) \phi(x)}{f(x) \phi(x)} \{ \log \phi(x) - \log f(x) \} dx \) is equal to
(a) \( \log \frac{\phi(x)}{f(x)} + k \)
(b) \( \frac{1}{2} \left\{ \log \frac{\phi(x)}{f(x)} \right\}^2 + k \)
(c) \( \frac{\phi(x)}{f(x)} \log \frac{\phi(x)}{f(x)} + k \)
(d) None of the options
Answer: (b) \( \frac{1}{2} \left\{ \log \frac{\phi(x)}{f(x)} \right\}^2 + k \)

Question. \( \int \sin 2x \cdot \log \cos x \, dx \) is equal to
(a) \( \cos^2 x \left( \frac{1}{2} + \log \cos x \right) + k \)
(b) \( \cos^2 x \cdot \log \cos x + k \)
(c) \( \cos^2 x \left( \frac{1}{2} - \log \cos x \right) + k \)
(d) None of the options
Answer: (c) \( \cos^2 x \left( \frac{1}{2} - \log \cos x \right) + k \)

Question. \( \int e^{-x}(1 - \tan x) \sec x \, dx \) is equal to
(a) \( e^{-x} \sec x + c \)
(b) \( e^{-x} \tan x + c \)
(c) \( -e^{-x} \tan x + c \)
(d) None of the options
Answer: (d) None of the options

Question. \( \int \frac{1 + \sin x}{1 + \cos x} \cdot e^x \, dx \) is equal to
(a) \( e^x \tan\left( \frac{x}{2} \right) + k \)
(b) \( e^x \tan x + k \)
(c) \( \frac{1}{2} e^x \tan \frac{x}{2} + k \)
(d) \( e^x \sec^2 \frac{x}{2} + k \)
Answer: (a) \( e^x \tan\left( \frac{x}{2} \right) + k \)

Question. Let \( \int e^x \{ f(x) - f'(x) \} dx = \phi(x) \). Then \( \int e^x f(x) \, dx \) is
(a) \( \phi(x) + e^x f(x) \)
(b) \( \phi(x) - e^x f(x) \)
(c) \( \frac{1}{2} \{ \phi(x) + e^x f(x) \} \)
(d) \( \frac{1}{2} \{ \phi(x) + e^x f'(x) \} \)
Answer: (c) \( \frac{1}{2} \{ \phi(x) + e^x f(x) \} \)

Question. If \( f(0) = f'(0) = 0 \) and \( f''(x) = \tan^2 x \) then \( f(x) \) is
(a) \( \log \sec x - \frac{1}{2} x^2 \)
(b) \( \log \cos x + \frac{1}{2} x^2 \)
(c) \( \log \sec x + \frac{1}{2} x^2 \)
(d) None of the options
Answer: (a) \( \log \sec x - \frac{1}{2} x^2 \)

Question. Let \( f(x) = \int \frac{x^2 dx}{(1 + x^2)(1 + \sqrt{1 + x^2})} \) and \( f(0) = 0 \). Then \( f(1) \) is
(a) \( \log(1 + \sqrt{2}) \)
(b) \( \log(1 + \sqrt{2}) - \frac{\pi}{4} \)
(c) \( \log(1 + \sqrt{2}) + \frac{\pi}{4} \)
(d) None of the options
Answer: (b) \( \log(1 + \sqrt{2}) - \frac{\pi}{4} \)

Question. \( \int \frac{dx}{\cos x + \sqrt{3} \sin x} \) is equal to
(a) \( \log \tan \left( \frac{x}{2} + \frac{\pi}{3} \right) + k \)
(b) \( \log \tan \left( \frac{x}{2} - \frac{\pi}{3} \right) + k \)
(c) \( \frac{1}{2} \log \tan \left( \frac{x}{2} + \frac{\pi}{3} \right) + k \)
(d) None of the options
Answer: (c) \( \frac{1}{2} \log \tan \left( \frac{x}{2} + \frac{\pi}{3} \right) + k \)

Choose the correct options. One or more options may be correct.

Question. If \( \int \tan^4 x \, dx = a \tan^3 x + b \tan x + \phi(x) \) then
(a) \( a = \frac{1}{3} \)
(b) \( b = 1 \)
(c) \( \phi(x) = x + c \)
(d) \( b = -1 \)
Answer: (a) \( a = \frac{1}{3} \) (c) \( \phi(x) = x + c \) (d) \( b = -1 \)

Question. If \( \int \frac{\sin x}{\sin(x - \alpha)} dx = Ax + B \log \sin(x - \alpha) + C \) then
(a) \( A = \sin \alpha \)
(b) \( B = \cos \alpha \)
(c) \( A = \cos \alpha \)
(d) \( B = \sin \alpha \)
Answer: (c) \( A = \cos \alpha \) (d) \( B = \sin \alpha \)

Question. If \( \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \log_e(9e^{2x} - 4) + C \) then
(a) \( A = \frac{3}{2} \)
(b) \( B = \frac{35}{36} \)
(c) C is indefinite
(d) \( A + B = -\frac{19}{36} \)
Answer: (b) \( B = \frac{35}{36} \) (c) C is indefinite (d) \( A + B = -\frac{19}{36} \)

Question. If \( \int x \log(1 + x^2) dx = \phi(x) \cdot \log(1 + x^2) + \psi(x) + c \) then
(a) \( \phi(x) = \frac{1 + x^2}{2} \)
(b) \( \psi(x) = \frac{1 + x^2}{2} \)
(c) \( \psi(x) = -\frac{1 + x^2}{2} \)
(d) \( \phi(x) = -\frac{1 + x^2}{2} \)
Answer: (a) \( \phi(x) = \frac{1 + x^2}{2} \) (c) \( \psi(x) = -\frac{1 + x^2}{2} \)

Question. \( \int \frac{dx}{(x + 1)(x - 2)} = A \log(x + 1) + B \log(x - 2) + C \), where
(a) \( A + B = 0 \)
(b) \( AB = -1 \)
(c) \( A : B = -1 \)
(d) None of the options
Answer: (a) \( A + B = 0 \) (c) \( A : B = -1 \)