Calculus

Integrals

Hard

core

Match List I with List II

LIST I LIST II
A. $\int \frac{\sin x}{1 + \cos x} \, dx$ I. $e^{\tan^{-1} x} + C$
B. $\int \frac{1}{1 - \tan x} \, dx$ II. $\log(\log x + 1) + C$
C. $\int \frac{e^{\tan^{-1} x}}{1 + x^2} \, dx$ III. $-\log\lvert 1+\cos x \rvert + C$
D. $\int \frac{1}{x + x \log x} \, dx$ IV. $\frac{x}{2} - \frac{1}{2}\log\lvert \cos x - \sin x \rvert + C$

Choose the correct answer from the options given below:

LIST I	LIST II
A. $\int \frac{\sin x}{1 + \cos x} \, dx$	I. $e^{\tan^{-1} x} + C$
B. $\int \frac{1}{1 - \tan x} \, dx$	II. $\log(\log x + 1) + C$
C. $\int \frac{e^{\tan^{-1} x}}{1 + x^2} \, dx$	III. $-\log\lvert 1+\cos x \rvert + C$
D. $\int \frac{1}{x + x \log x} \, dx$	IV. $\frac{x}{2} - \frac{1}{2}\log\lvert \cos x - \sin x \rvert + C$

A-II, B-III, C-IV, D-I

A-III, B-IV, C-I, D-II

A-I, B-II, C-III, D-IV

A-IV, B-I, C-III, D-II

✅ Correct Option: 2

A: Let $u = 1+\cos x$ , $du = -\sin x\,dx$ . Result: $-\log|1+\cos x| + C$ (III).

B: Write as $\frac{\cos x}{\cos x - \sin x}$ , split. Result: $\frac{x}{2} - \frac{1}{2}\log|\cos x - \sin x| + C$ (IV).

C: Let $u = \tan^{-1}x$ . Result: $e^{\tan^{-1}x} + c$ (I).

D: Let $u = 1+\log x$ . Result: $\log|1+\log x| + C$ (II).

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The value of integral $\int \sqrt{4x^2 + 9}\, dx$ is

Solution