Integrals — Class 12 Mathematics PYQs

${\int }_{-1}^1\frac{|x|}{x}dx$, $x\ne 0$ is equal to

(A) $-1$
(B) $0$
(C) $1$
(D) $2$

If $\int \frac{2^x}{x^2}dx=k\cdot 2^{1/x}+C$, then $k$ is equal to

(A) $\frac{-1}{\log 2}$
(B) $-\log 2$
(C) $-1$
(D) $\frac{1}{2}$

$\int \frac{1-2\sin x}{{\cos }^{2}x}dx$ is equal to :

(A) $\tan x-2\sec x+C$
(B) $-\tan x+2\sec x+C$
(C) $-\tan x-2\sec x+C$
(D) $\tan x+2\sec x+C$

If ${\int }_0^1\frac{e^x}{1+x}dx=\alpha$, then ${\int }_0^1\frac{e^x}{(1+x{)}^2}dx$ is equal to

(A) $\alpha -1+\frac{e}{2}$
(B) $\alpha +1-\frac{e}{2}$
(C) $\alpha -1-\frac{e}{2}$
(D) $\alpha +1+\frac{e}{2}$

If $f(2a-x)=f(x)$, then ${\int }_0^{2a}f(x)dx$ is

(A) ${\int }_0^{2a}f\left(\frac{x}{2}\right)dx$
(B) ${\int }_0^{a}f(x)dx$
(C) $2{\int }_0^{a}f(x)dx$
(D) $2{\int }_a^{0}f(x)dx$

$\int \frac{dx}{{\sin }^{2}x{\cos }^{2}x}$ is equal to

(A) $\tan x+\cot x+C$
(B) $(\tan x+\cot x{)}^2+C$
(C) $\tan x-\cot x+C$
(D) $(\tan x-\cot x{)}^2+C$

If ${\int }_0^{a}xdx\le \frac{a}{2}+6$, then

(A) $-4\le a\le 3$
(B) $a\ge 4$, $a\le -3$
(C) $-3\le a\le 4$
(D) $-3\le a\le 0$

Evaluate : ${\int }_0^{\pi /4}\sqrt{1+\sin 2x}dx$

(a) Find : $\int \frac{x+\sin x}{1+\cos x}dx$

OR

(b) Evaluate : ${\int }_0^{\pi /4}\frac{dx}{{\cos }^{3}x\sqrt{2\sin 2x}}$

(a) Find : $\int \frac{\cos 2x}{(\sin x+\cos x{)}^2}dx$

OR

(b) Evaluate : ${\int }_0^{\pi /2}\frac{5\sin x+3\cos x}{\sin x+\cos x}dx$