Chapter Questions
The equation of a curve passing through origin is given by $y=\int x^{3} \cos x^{4} d x$. If the equation of the curve is written in the form $x=g(y)$, then(A) $g(y)=\sqrt[3]{\sin ^{-1}(4 y)}$(B) $g(y)=\sqrt{\sin ^{-1}(4 y)}$(C) $g(y)=\sqrt[4]{\sin ^{-1}(4 y)}$(D) none of these
If $\phi(x)=\int \frac{d x}{\sin ^{12} x \cos ^{7 / 2} x^{2}}$, then $\phi\left(\frac{\pi}{4}\right)-\phi(0)=$(A) $\frac{12}{5}$(B) $\frac{9}{5}$(C) $\frac{6}{5}$(D) 0
If $\phi(x)=\lim _{n \rightarrow \infty} \frac{x^{n}-x^{-11}}{x^{n}+x^{-n}}, 0<x<1, n \in N$, then$\int \sin ^{-1} x \phi(x) d x$ is equal to(A) $x \sin ^{-1} x+\sqrt{1-x^{2}}+C$(B) $-\left(x \sin ^{-1} x+\sqrt{1-x^{2}}\right)+C$(C) $x \sin ^{-1} x-\sqrt{1-x^{2}}+C$(D) none of these.
$\int \frac{\sin ^{8} x-\cos ^{8} x}{1-2 \sin ^{2} x \cos ^{2} x} d x$ is equal to(A) $\frac{1}{2} \sin 2 x+c$(B) $-\frac{1}{2} \sin 2 x+c$(C) $-\frac{1}{2} \sin x+c$(D) $\quad-\sin ^{2} x+c$
If $\int \frac{(\sqrt{x})^{5}}{(\sqrt{x})^{7}+x^{6}} d x=a \ln \left(\frac{x^{k}}{x^{k}+1}\right)+c$, the values of$a$ and $k$ respectively are(A) $\frac{5}{2}$ and $\frac{2}{5}$(B) $\frac{2}{5}$ and $\frac{5}{2}$(C) $\frac{5}{2}$ and 2(D) none of there
$\int x\left[f\left(x^{2}\right) g^{\prime \prime}\left(x^{2}\right)-f^{\prime \prime}\left(x^{2}\right) g\left(x^{2}\right)\right] d x$(A) $f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)+c$(B) $\frac{1}{2}\left[f\left(x^{2}\right) g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right]+c$(C) $\frac{1}{2}\left[f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right]+c$(D) none of the above
The anti-derivative of $\frac{\cos 5 x+\cos 4 x}{1-2 \cos 3 x}$ is(A) $\frac{\sin 2 x}{2}+\cos x+c$(B) $-\frac{\sin 2 x}{2}+\sin x+c$(C) $-\frac{\sin 2 x}{2}-\sin x+c$(D) $\frac{\sin 2 x}{2}-\cos x+c$
If $\int \tan ^{4} x d x n=K \tan ^{3} x+L \tan x+f(x)$, then(A) $K=\frac{1}{3}, L=-1, f(x)=x+C$(B) $K=1, L=-1, f(x)=-x+C$(C) $K=-1, L=1, f(x)=2 x+C$(D) $K=\frac{1}{2}, L=\frac{1}{3}, f(x)=3 x+C$
$\int \frac{1}{\left[(x-1)^{3}(x+2)^{5}\right]^{1 / 4}} d x$ is equal to(A) $\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{1 / 4}+c$(B) $\frac{4}{3}\left(\frac{x+2}{x-1}\right)^{14}+c$(C) $\frac{1}{3}\left(\frac{x-1}{x+2}\right)^{1 / 4}+c$(D) $\frac{1}{3}\left(\frac{x+2}{x-1}\right)^{1 / 4}+c$
$$\int\left(\frac{\ln x-1}{(\ln x)^{2}+1}\right)^{2} d x \text { is equal to }$$(A) $\frac{x}{x^{2}+1}+c$(B) $\frac{\ln x}{(\ln x)^{2}+1}+c$(C) $\frac{x}{(\ln x)^{2}+1}+c$(D) $e^{x}\left(\frac{x}{x^{2}+1}\right)+c$
The value of $\int \frac{a x^{2}-b}{x \sqrt{c^{2} x^{2}-\left(a x^{2}+b\right)^{2}}} d x$ is equal to(A) $\sin ^{-1}\left(\frac{\left(a x+\frac{b}{x}\right)}{c} \mid+k\right.$(B) $\sin ^{-1}\left(\frac{a x^{2}+\frac{b}{x^{2}}}{c}\right)+k$(C) $\cos ^{-1}\left(\frac{a x+\frac{b}{x}}{c}\right)+k$(d) $\cos ^{-1}\left|\frac{\left(a x^{2}+\frac{b}{x^{2}}\right)}{c}\right|+k$
The value of $\int \frac{\sec x d x}{\sqrt{\sin (2 x+\theta)+\sin \theta}}$ is(A) $\sqrt{(\tan x+\tan \theta) \sec \theta}+c$(B) $\sqrt{2(\tan x+\tan \theta) \sec \theta}+c$(C) $\sqrt{2(\sin x+\tan \theta) \sec \theta}+c$(D) none of these
$\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x$ is equal to(A) $\sin ^{-1}(\sin x+\cos x)+c$(B) $\sin ^{-1}\left[\frac{1}{3}(\sin x+\cos x)\right]+c$(C) $\cos ^{-1}(\sin x+\cos x)+c$(D) none of these
If $f(x)=\int \frac{x^{2} d x}{\left(1+x^{2}\right)\left(1+\sqrt{1+x^{2}}\right)}$ and $f(0)=0$, thenthe value of $f(1)$ is(A) $\log (1+\sqrt{2})$(B) $\log (1+\sqrt{2})-\frac{\pi}{4}$(C) $\log (1+\sqrt{2})+\frac{\pi}{2}$(D) none of these
If $\int \frac{(x+1)}{x\left(1+x e^{x}\right)^{2}} d x=\log |1-f(x)|+f(x)+C$, then $f(x)=$(A) $\frac{1}{x+e^{x}}$(B) $\frac{1}{1+x e^{x}}$(C) $\frac{1}{\left(1+x e^{x}\right)^{2}}$(D) $\frac{1}{\left(x+e^{x}\right)^{2}}$
If $\int f(x) \sin x \cos x d x=\frac{1}{2\left(b^{2}-a^{2}\right)} \log [f(x)]+C$,then $f(x)$ is equal to(A) $\frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$(B) $\frac{1}{a^{2} \sin ^{2} x-b^{2} \cos ^{2} x}$(C) $\frac{1}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}$(D) $\frac{1}{a^{2} \cos ^{2} x-b^{2} \sin ^{2} x}$
$\int \frac{d x}{(x+a)^{N 7}(x-b)^{67}}$ is equal to(A) $\left(\frac{7}{a+b}\right)\left(\frac{x+a}{x-b}\right)^{17}+c$(B) $\left(\frac{7}{a+b}\right)\left(\frac{x-b}{x+a}\right)^{17}+c$(C) $\frac{6}{a+b}\left(\frac{x-b}{x+a}\right)^{16}+c$(D) $\frac{6}{a+b}\left(\frac{x+a}{x-b}\right)^{16}+c$
$\int \sin (\log x) d x=f(x)[\sin g(x)-\cos h(x)]+c$, then(A) $\lim _{x \rightarrow 2} f(x)=2$(B) $g\left(e^{3}\right)=-3$(C) $h\left(e^{5}\right)=-4$(D) $\lim _{x \rightarrow 1} \frac{g(x)}{h(x)}=1$
$$\int \frac{\sqrt{x}}{\sqrt{x^{3}+4}} d x \text { equals }$$(A) $\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C$(B) $\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C$(C) $\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C$(D) none of these
$\int \frac{[f(x), \phi(x)-\phi(x), \phi(x)]}{f(x) \cdot \phi(x)} \log \frac{f(x)}{\phi(x)} d x$ is equal to(A) $\log \frac{\phi(x)}{f(x)}+k$(B) $\left.\frac{1}{2} \mid \log \frac{\phi(x)}{f(x)}\right]^{2}+k$(C) $\frac{\phi(x)}{f(x)} \log \frac{\phi(x)}{f(x)}+k$(D) none of these
If $I=\int \frac{1}{2 p} \sqrt{\frac{p-1}{p+1}} d p=f(p)+c$, then $f(p)$ is equal to(A) $\frac{1}{2} \ln \left(p-\sqrt{p^{2}-1}\right)$(B) $\left(\frac{1}{2} \cos ^{-1} p+\frac{1}{2} \sec ^{-1} p\right)$(C) $\frac{1}{2} \ln \sqrt{p+\sqrt{p^{2}-1}}-\frac{1}{2} \sec ^{-1} p$(D) none of these
If $\int \frac{1}{x+x^{5}} d x=f(x)+c$, then $\int \frac{x^{4}}{x+x^{5}} d x$ is equal to(A) $\log |x|+f(x)+c$(B) $\log |x|-f(x)+c$(C) $x f(x)+c$(D) none of these
If $l^{\prime}(x)$ means $\log \log \log \ldots x$, the log being repeated $r$times, then $\int\left[x /(x) l^{2}(x) l^{3}(x) \ldots l^{\prime}(x)\right]^{-1} d x$ is equal to(A) $l^{-1}(x)+C$(B) $\frac{l^{r+1}(x)}{r+1}+C$(C) $l^{\prime}(x)+C$(D) none of these
$\int \frac{\left(x^{2}-2\right) d x}{\left(x^{4}+5 x^{2}+4\right) \tan ^{-1}\left(\frac{x^{2}+2}{x}\right)}$ is(A) $\log \left|\tan ^{-1} \sqrt{x+2}\right|+C$(B) $\log \left|\tan ^{-1}\left(x+\frac{2}{x}\right)\right|+C$.(C) $\sin ^{-1}\left(\frac{x+2}{x}\right)+C$(D) $\tan ^{-1}\left(\frac{x+2}{x}\right)+C$
The value of $\int e^{x} \frac{1+n x^{n-1}-x^{2 n}}{\left(1-x^{n}\right) \sqrt{1-x^{2 n}}} d x$ is(A) $e^{x} \frac{\sqrt{1-x^{n}}}{1-x^{n}}+C$(B) $e^{x} \frac{\sqrt{1+x^{2 n}}}{1-x^{2 n}}+C$(C) $e^{x} \frac{\sqrt{1-x^{2 n}}}{1-x^{2 n}}+C$(D) $e^{x} \frac{\sqrt{1-x^{2 n}}}{1-x^{\prime \prime}}+C$
$\int \sqrt{\frac{\cos x-\cos ^{3} x}{1-\cos ^{3} x}} d x$ is equal to(A) $\frac{2}{3} \sin ^{-1}\left(\cos ^{1 / 2} x\right)+c$(B) $\frac{2}{3} \sin ^{-1}\left(\cos ^{1 / 2} x\right)+$(C) $\frac{2}{3} \cos ^{-1}\left(\cos ^{1 / 2} x\right)+c$(D) none of the above
$\int \cos ^{-17} x \sin ^{-117} x d x=$(A) $\log \left|\sin ^{4 / 7} x\right|+c$(B) $\frac{4}{7} \tan ^{4 \pi} x+c$(C) $\frac{-7}{4} \tan ^{-47} x+c$(D) $\log \left|\cos ^{37} x\right|+c$(e) $\frac{7}{4} \tan ^{-4 / 7} x+c$
If the integral $\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|$$+k$, then $a$ is equal to:(A) $-1$(B) $-2$(C) 1(D) 2
The integral $\int \frac{d x}{\left(a^{2}-b^{2} x^{2}\right)^{3 / 2}}$, equals:(A) $\frac{x}{\sqrt{a^{2}-b^{2} x^{2}}}+C$(B) $\frac{x}{a^{2} \sqrt{a^{2}-b^{2} x^{2}}}+C$(C) $\frac{a x}{\sqrt{a^{2}-b^{2} x^{2}}}+C$(D) $\frac{x}{a^{2} \sqrt{a^{2}-b^{2} x^{2}}}+C$
The value of $\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}$ is(A) $x+\ln \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c$(B) $x-\ln \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c$(C) $x+\ln \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c$(D) $x-\ln \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c$
For a natural number $n$, the value of the integral$\int\left(x^{3 n}+x^{2 n}+x^{n}\right)\left(2 x^{2 n}+3 x^{\prime \prime}+6\right)^{1 / 1 \prime} d x$ is(A) $\frac{1}{6 n}\left(2 x^{3 n}+3 x^{2 n}+6 x^{n}\right)^{1 / n}+C$(B) $\frac{1}{6 n}\left(2 x^{3 n}+3 x^{2 n}+6 x^{n}\right)^{1 / n+1}+C$(C) $\frac{1}{6(n+1)}\left(2 x^{3 n}+3 x^{2 n}+6 x^{n}\right)^{1 / n+1}+C$(D) none of these
If $f\left(\frac{3 x-4}{3 x+4}\right)=x+2$, then $\int f(x) d x$ is equal to(A) $\operatorname{ex}-{ }^{2} \ln \left|\frac{3 x-4}{3 x+4}\right|+c$(B) $-\frac{8}{3} \ln |x-1|+\frac{2}{3} x+c$(C) $\frac{8}{3} \ln |x-1|+\frac{x}{3}+c$(D) none of these
$\int \frac{\sqrt{1+\sqrt[3]{x}}}{\sqrt[3]{x^{2}}} d x$ is equal to(A) $\left(1+x^{1 / 3}\right)^{3 / 2}+C$(B) $-\left(1+x^{1 / 3}\right)^{1 / 2}+C$(C) $2\left(1+x^{1 / 3}\right)^{3 / 2}+C$(D) none of these
$\int x\left\{f\left(x^{2}\right) g^{\prime \prime}\left(x^{2}\right)-f^{\prime \prime}\left(x^{2}\right) g\left(x^{2}\right)\right\} d x=$(A) $f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)+c$(B) $\frac{1}{2}\left\{f\left(x^{2}\right) g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right\}+c$(C) $\frac{1}{2}\left\{f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right\}+c$(D) none of these
(A) $\frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$(B) $\frac{1}{a^{2} \sin ^{2} x-b^{2} \cos ^{2} x}$(C) $\frac{1}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}$(D) $\frac{1}{a^{2} \cos ^{2} x-b^{2} \sin ^{2} x}$
If $f(x)=\tan ^{-1} x+\ln \sqrt{1+x}-\ln \sqrt{1-x}$, then theintegral of $\frac{1}{2} f^{\prime}(x)$ w.r.t. $x^{4}$ is(A) $\ln \left|1-x^{4}\right|+C$(B) $-\ln \left|1-x^{4}\right|+C$(C) $\ln \left|x^{4}-1\right|+C$(D) $\ln \left|1+x^{4}\right|+C$
$\int \frac{e^{x}\left(2-x^{2}\right)}{(1-x) \sqrt{1-x^{2}}} d x=$(A) $e^{x} \frac{\sqrt{1+x}}{\sqrt{1-x^{2}}}+C$(B) $e^{x} \frac{\sqrt{1-x}}{\sqrt{1+x}}+C$(C) $e^{x} \frac{\sqrt{1+x}}{\sqrt{1-x}}+C$(D) none of these
$\int\left\{\left(\frac{x}{e}\right)^{x}+\left(\frac{e}{x}\right)^{x}\right\} \ln x d x=$(A) $\left(\frac{x}{e}\right)^{x}-\left(\frac{e}{x}\right)^{x}+C$(B) $\left(\frac{x}{e}\right)^{x}+\left(\frac{e}{x}\right)^{x}+C$(C) $\left(\frac{x}{e}\right)^{x}-2\left(\frac{e}{x}\right)^{x}+C$(D) none of these
$\int\left(x^{3} a+x^{2} a+x a\right)\left(2 x^{2} a+3 x a+6\right)^{1} a d x=$(A) $\frac{1}{6(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1-\frac{1}{4}}+C$(B) $\frac{1}{6(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1+\frac{1}{a}}+C$(C) $\frac{1}{3(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1+\frac{1}{a}}+C$(D) none of these
If $y(x-y)^{2}=x$, then $\int \frac{d x}{x-3 y}=$(A) $\frac{1}{2} \ln \left|(x-y)^{2}+1\right|+C$(B) $\frac{1}{2} \ln \left|(x-y)^{2}-1\right|+C$(C) $\frac{1}{4} \ln \left|(x-y)^{2}+1\right|+C$(D) none of these.
$\int \frac{\cos x}{1-\sin x \cos x} d x=\tan ^{-1}(\sin x-\cos x)$$+\frac{k}{\sqrt{3}} \ln \left|\frac{\sin x+\cos x-\sqrt{3}}{\sin x+\cos x-\sqrt{3}}\right|+C$, where $k=$(A) $-\frac{1}{x}$(B) $\frac{1}{2}$(C) $-1$(D) 1
$\int \frac{\cos \left(x+\frac{\pi}{4}\right)}{2+\sin 2 x} d x$(A) $\sqrt{2} \tan ^{-1}(\sin x-\cos x)+C$(B) $\frac{1}{\sqrt{2}} \tan ^{-1}(\sin x-\cos x)+C$(C) $\frac{1}{\sqrt{2}} \tan ^{-1}(\sin x+\cos x)+C$(D) $\sqrt{2} \tan ^{-1}(\sin x+\cos x)+C$
$\int \frac{d x}{1-\cos ^{4} x}=-\frac{1}{2 \tan x}+\frac{k}{\sqrt{2}} \tan ^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+C$,where $k=$(A) $\frac{1}{2}$(B) $-\frac{1}{2}$(C) $-1$(D) 1
$\int \frac{\sqrt{\cot x}-\sqrt{\tan x}}{1+3 \sin 2 x} d x$$=k \tan ^{-1}\left(\frac{\sqrt{\tan x}+\sqrt{\cot x}}{2}\right)+C$where $k=$(A) 1(B) $-1$(C) 2(D) $-2$
$\int \frac{\cos 5 x+\cos 4 x}{1-2 \cos 3 x} d x=k \sin x(1+\cos x)+C$, where$k=$(A) 2(B) $-2$(C) 1(D) $-1$
$\int \frac{\sec x d x}{\sqrt{\sin (2 x+a)+\sin a}}=k \sqrt{\tan x+\tan a}+C$,where $k=$(A) $\sqrt{\frac{2}{\cos a}}$(B) $\sqrt{2 \cos a}$(C) $\sqrt{\cos a}$(D) $\sqrt{\frac{1}{\cos a}}$
$\sqrt{x+\sqrt{x^{2}+2}} d x$$=\frac{1}{3}\left(\sqrt{x^{2}+2}+x\right)^{3 / 2}+k\left(\sqrt{x^{2}+2}-x\right)^{1 / 2}+C$,where $k=$(A) 2(B) $\sqrt{2}$(C) $-2$(D) $-\sqrt{2}$
$\int \frac{x^{4}-1}{x^{2} \sqrt{x^{4}+x^{2}+1}} d x=$(A) $\frac{\sqrt{x^{4}+x^{2}+1}}{x}+C$(B) $\frac{x}{\sqrt{x^{4}+x^{2}+1}}+C$(C) $-\frac{\sqrt{x^{4}+x^{2}+1}}{x}+C$(D) none of these
$\int \frac{x^{2}-1}{\left(x^{2}+1\right) \sqrt{1+x^{4}}} d x=k \cos ^{-1}\left(\frac{\sqrt{2} x}{x^{2}+1}\right)+C$,where $k=$(A) $\frac{1}{2}$(B) 2(C) $\frac{1}{\sqrt{2}}$(D) $\sqrt{2}$
$\int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^{2}}}=k\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right)+C$, where $k=$(A) 1(B) 2(C) 3(D) 4
$\int \frac{d x}{\cos ^{3} x \sqrt{\sin 2 x}}=$(A) $\sqrt{2}\left(\tan ^{1 / 2} x+\frac{1}{5} \tan ^{5 / 2} x\right)+C$(B) $\sqrt{2}\left(\cot ^{1 / 2} x+\frac{1}{5} \cot ^{5 / 2} x\right)+C$(C) $\sqrt{2}\left(\tan ^{1 / 2} x-\frac{1}{5} \tan ^{5 / 2} x\right)+C$(D) none of these
$\int \frac{1+x^{4}}{\left(1-x^{4}\right)^{3 / 2}} d x=$(A) $\frac{1}{\sqrt{x^{2}-\frac{1}{x^{2}}}}+c$(B) $\frac{1}{\sqrt{\frac{1}{x^{2}}-x^{2}}}+c$(C) $\frac{1}{\sqrt{x^{2}+\frac{1}{x^{2}}}}+c$(D) none of these
$\int \frac{\left(x^{2}-1\right)}{\left(x^{4}+3 x^{2}+1\right) \tan ^{-1}\left(x+\frac{1}{x}\right)} d x$(A) $\log \left|\tan ^{-1}\left(x+\frac{1}{x}\right)\right|+c$(B) $\log \left|\cot ^{-1}\left(x+\frac{1}{x}\right)\right|+c$(C) $2 \log \left|\tan ^{-1}\left(x+\frac{1}{x}\right)\right|+c$(D) none of these
$\int \sqrt{\frac{\cos x-\cos ^{3} x}{1-\cos ^{3} x}} d x=$(A) $\frac{2}{3} \sin ^{-1}\left(\cos ^{32} x\right)+c$(B) $-\frac{2}{3} \sin ^{-1}\left(\cos ^{32} x\right)+c$(C) $\frac{3}{2} \sin ^{-1}\left(\cos ^{32} x\right)+c$(D) none of these
$\int \frac{d x}{(x-1)^{3 / 4}(x+2)^{5 / 4}}=$(A) $\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{14}+c$(B) $\frac{3}{4}\left(\frac{x-1}{x+2}\right)^{14}+c$(C) $\frac{4}{3}\left(\frac{x+2}{x-1}\right)^{14}+c$(D) none of these
$\int \frac{\cos 7 x-\cos 8 x}{1+2 \cos 5 x} d x=$(A) $\frac{\sin 2 x}{2}+\frac{\sin 3 x}{3}+c$(B) $-\frac{\sin 2 x}{2}-\frac{\sin 3 x}{3}+c$(C) $\frac{\sin 2 x}{2}-\frac{\sin 3 x}{3}+c$(D) none of these.
If $I n=\int x^{n} \sqrt{a^{2}-x^{2}} d x$, then $(n+2) \ln -(n-1) a^{2} I n$$-2=$(A) $x^{n} \sqrt{a^{2}-x^{2}}$(B) $-x^{n-1} \sqrt{a^{2}-x^{2}}$(C) $-x n^{-1}\left(a^{2}-x^{2}\right)^{1 / 2}$(D) none of these
$\int \frac{\left(1-\cot ^{\prime-2} x\right) d x}{\tan x+\cot x \cdot \cot ^{n-2} x}=$(A) $\frac{1}{n} \log \left|\sin ^{\prime \prime} x-\cos ^{n} x\right|+c$(B) $\frac{1}{n} \log \left|\sin ^{n} x+\cos ^{n} x\right|+c$(C) $\frac{1}{n-1} \log \left|\sin ^{n} x+\cos ^{\prime \prime} x\right|+c$(D) none of these
$\int e^{\operatorname{lan} x}(\sec x-\sin x) d x=$(A) $e^{\ln x \sin x+c}$(B) $-e^{\ln x} x \sin x+c$(C) $-e^{\ln x} x \cos x+c$(D) $e^{\operatorname{lan}} x \cos x+c$
$\int \frac{(x-1) d x}{(x+1) \sqrt{x^{3}+x^{2}+x}}=k \tan ^{-1} \sqrt{\frac{x^{2}+x+1}{x}}+c^{\prime}$where $k=$(A) $]$(B) 2(C) 4(D) none of these
$\int \frac{d x}{\tan x+\cot x+\sec x+\operatorname{cosec} x}=$(A) $\frac{1}{2}(\sin x-\cos x+x)+c$(B) $\frac{1}{2}(\sin x-\cos x-x)+c$(C) $\frac{1}{2}(\sin x+\cos x+x)+c$(D) none of these
If $\int f(x) \sin x \cos x d x=\frac{1}{2\left(b^{2}-a^{2}\right)} \log f(x)+c$then $f(x)$ is equal to(A) $\frac{1}{a^{2} \sin ^{2} x-b^{2} \cos ^{2} x}$(B) $\frac{1}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}$(C) $\frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$(D) none of these
$\int \frac{(\sin x-\cos x) d x}{(\sin x+\cos x) \sqrt{\sin x \cos x+\sin ^{2} x \cos ^{2} x}}=$(A) $\operatorname{cosec}^{-1}(1+\sin 2 x)+c$(B) $-\operatorname{cosec}^{-1}(1+\sin 2 x)+c$(C) $\sec ^{-1}(1+\sin 2 x)+c$(D) $-\sec ^{-1}(1+\sin 2 x)+c$
$\int \frac{d x}{\left(2 a x+x^{2}\right)^{1 / 2}}=$(A) $\frac{1}{a^{2}} \frac{x+a}{\sqrt{x^{2}+2 a x}}+c$(B) $\frac{1}{a^{2}} \frac{x-a}{\sqrt{x^{2}+2 a x}}+c$(C) $-\frac{1}{a^{2}} \cdot \frac{x+a}{\sqrt{x^{2}+2 a x}}+c$(D) none of these
$\int \frac{\sin ^{3} \theta / 2}{\cos \theta / 2 \sqrt{\cos ^{3} \theta+\cos ^{2} \theta+\cos \theta}} d \theta$$=\tan ^{-1} \sqrt{k}+C$, where $k=$(A) $\cos \theta+\sec \theta+1$(B) $\cos \theta-\sec \theta+1$(C) $\cos \theta+\sec \theta-1$(D) none of these
$\int \frac{(2 \sin \theta+\sin 2 \theta) d \theta}{(\cos \theta-1) \sqrt{\cos \theta+\cos ^{2} \theta+\cos ^{3} \theta}}$$=-\frac{2}{3} \log \left|\frac{k-\sqrt{3}}{k+\sqrt{3}}\right|+c$, where $k=$(A) $\sqrt{\cos \theta+\sec \theta+1}$(B) $\cos \theta+\sec \theta+1$(C) $\sqrt{\cos \theta+\sec \theta-1}$(D) $\cos \theta+\sec \theta-1$
If $[\cdot]$ and $\{\cdot\}$ denote the greatest integer and fractional part, respectively, then $\int[\{[\{[x]\}]\}] d x$ is equal to(A) $C_{r^{2}}$(B) $x+C$(C) $\frac{x^{2}}{2}+C$(D) cannot be integrated
If $f(x)=\lim _{n \rightarrow \infty}\left[2 x+4 x^{3}+\ldots+2 n x^{2 n-1}\right], 0<x<1$,then $\int f(x) d x=$(A) $\frac{1}{\sqrt{1-x^{2}}}+C$(B) $\sqrt{1-x^{2}}+C$(C) $\left(1-x^{2}\right)+C$(D) $\frac{1}{1-x^{2}}+C$
If $I m, n=\int \cos ^{\mathrm{m}} x \sin n x d x$, then $7 I_{4,3}-4 I_{3,2}=$(A) $-\cos 3 x \cos ^{4} x+C$(B) $\cos 3 x \cos ^{4} x+C$(C) $-\frac{1}{3} \cos 3 x \cos ^{4} x+C$(D) none of these
$\int \frac{d x}{(1+\sqrt{x})^{8}}=\frac{-1}{3(1+\sqrt{x})^{k_{1}}}+\frac{2}{7(1+\sqrt{x})^{k_{3}}}+C$where(A) $k_{1}=6$(B) $k_{2}=7$(C) $k_{1}=-6$(D) $k_{2}=-7$
If $\int f(x) \sin 2 x d x=\frac{\ln |f(x)|}{b^{2}-a^{2}}+C$, then $f(x)$ is equal(A) $\frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$(B) $\frac{1}{a^{2} \sin ^{2} x-b^{2} \cos ^{2} x}$(C) $\frac{1}{a^{2} \cos ^{2} x-b^{2} \sin ^{2} x}$(D) $\frac{1}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}$
If $\int \frac{(\sqrt{x})^{5}}{(\sqrt{x})^{7}+x^{6}} d x=a \log \left(\frac{x^{k}}{1+x^{k}}\right)+C$, then(A) $a=\frac{2}{5}$(B) $a=-\frac{2}{5}$(C) $k=\frac{5}{2}$(D) $k=-\frac{5}{2}$
If $P=\int e^{a x} \cos b x d x$ and $Q=\int e^{\alpha x} \sin b x d x$, then(A) $P=\frac{e^{a x}}{\sqrt{a^{2}+b^{2}}} \cos \left(b x-\tan ^{-1} \frac{b}{a}\right)$(B) $Q=\frac{e^{a x}}{\sqrt{a^{2}+b^{2}}} \sin \left(b x-\tan ^{-1} \frac{b}{a}\right)$(C) $\left(P^{2}+Q^{2}\right)\left(a^{2}+b^{2}\right)=e^{2} a x$(D) $\tan ^{-1}\left(\frac{Q}{P}\right)+\tan ^{-1} \frac{b}{a}=b x$
$\int \frac{x^{4}+1}{x^{6}+1} d x=\tan ^{-1} k_{1}-\frac{2}{3} \tan ^{-1} k_{2}+C$, where(A) $k_{1}=x+\frac{1}{x}$(B) $k_{2}=x^{3}$(C) $k_{1}=x-\frac{1}{v}$(D) $k_{2}=x^{4}$
If $\int \frac{x \log \left(x+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} d x$$=A \sqrt{1+x^{2}} \log \left(x+\sqrt{1+x^{2}}\right)+B x+C$, then(A) $A=-1$(B) $B=-1$(C) $A=1$(D) none of these
If $\int \frac{3 \cot 3 x-\cot x}{\tan x-3 \tan 3 x} d x=A x+B \log \left|\frac{\sqrt{3}-\tan x}{\sqrt{3}+\tan x}\right|+$$C$, then(A) $A=1$(B) $B=-\sqrt{3}$(C) $B=-\frac{1}{\sqrt{3}}$(D) none of these
Let $f(x)=\frac{x+2}{2 x+3}$, if $\int\left(\frac{f(x)}{x^{2}}\right)^{1 / 2} d x$$=\frac{1}{\sqrt{2}} g\left(\frac{1+\sqrt{2 f(x)}}{1-\sqrt{2 f(x)}}\right)-\sqrt{\frac{2}{3}} h\left(\frac{\sqrt{3 f(x)}+\sqrt{2}}{\sqrt{3 f(x)-\sqrt{2}}}\right)+C$then(A) $g(x)=\log |x|$(B) $h(x)=\log |x|$(C) $g(x)=\tan ^{-1} x$(D) $h(x)=\tan ^{-1} x$
If $f(x)=\lim _{n \rightarrow \infty} e^{x \tan \left(\frac{1}{n}\right) \log \left(\frac{1}{*}\right)}$ and $\int \frac{f(x)}{\sqrt[3]{\sin ^{11} x \cos x}} d x=$$g(x)+C$, then(A) $g\left(\frac{\pi}{4}\right)=\frac{15}{8}$(B) $g(x)$ is continuous for all $x$(C) $g\left(\frac{\pi}{4}\right)=-\frac{15}{8}$(D) $g(x)$ is not differentiable at infinitely many points
If for all $x \in[-1,0), \int\left(\cos ^{-1} x+\cos ^{-1} \sqrt{1-x^{2}}\right) d x$$=A x+f(x) \sin ^{-1} x-2 \sqrt{1-x^{2}}+C$, then(A) $A=\frac{\pi}{4}$(B) $A=\frac{\pi}{2}$(C) $f(x)=x$(D) $f(x)=-2 x$
If $\int \frac{x^{2}+20}{(x \sin x+5 \cos x)^{2}} d x=-\frac{x}{A \cos x}+B$, then(A) $A=x \sin x+5 \cos x$(B) $B=\cot x$(C) $A=-(x \sin x+5 \cos x)$(D) $B=\tan x$
$\int x^{1 / 3}\left(2+x^{2 / 3}\right)^{1 / 4} d x$ is equal to(A) $\frac{2}{3}\left(2+x^{2 / 3}\right)^{9 / 4}+\frac{12}{5}\left(2+x^{2 / 3}\right)^{5 / 4}+C$(B) $\frac{2}{3}\left(2+x^{2 / 3}\right)^{9 / 4}-\frac{12}{5}\left(2+x^{2 / 3}\right)^{5 / 4}+C$(C) $\frac{1}{3}\left(2+x^{2 / 3}\right)^{9 / 4}-\frac{12}{5}\left(2+x^{2 / 3}\right)^{5 / 4}+C$(D) none of these
$\int \frac{\sqrt{1+\sqrt{x}}}{x} d x$ is equal to(A) $2 \sqrt{1+\sqrt{x}}-2 \log \left(\frac{\sqrt{1+\sqrt{x}}-1}{\sqrt{1+\sqrt{x}}+1}\right)+C$(B) $4 \sqrt{1+\sqrt{x}}+2 \log \left(\frac{\sqrt{1+\sqrt{x}}+1}{\sqrt{1+\sqrt{x}}-1}\right)+C$(C) $\sqrt[4]{1+\sqrt{x}}+2 \log \left(\frac{\sqrt{1+\sqrt{x}}-1}{\sqrt{1+\sqrt{x}}+1}\right)+C$(D) none of these
$\int \frac{\sqrt[3]{1+\sqrt[4]{x}}}{\sqrt{x}} d x$ is equal to(A) $12\left(\frac{(1+\sqrt[4]{x})^{n / 3}}{7}+\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C$(B) $12\left(\frac{(1+\sqrt[4]{x})^{7 / 3}}{7}-\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C$(C) $6\left(\frac{(1+\sqrt[4]{x})^{7 / 3}}{7}-\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C$(D) none of these
$\int \sqrt[3]{x} \sqrt[7]{1+\sqrt[3]{x^{4}}} d x$ is equal to(A) $\frac{21}{32}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C$(B) $\frac{32}{21}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C$(C) $\frac{7}{32}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C$(D) none of these
If $I n=\int \tan n x d x$, then $I_{0}+I_{1}+2\left(I_{2}+\ldots+I_{n}\right)+I_{0}+I_{10}$is equal to(A) $\left(\frac{\tan x}{1}+\frac{\tan ^{2} x}{2}+\ldots+\frac{\tan ^{9} x}{9}\right)$(B) $-\left(\frac{\tan x}{1}+\frac{\tan ^{2} x}{2}+\ldots+\frac{\tan ^{9} x}{9}\right)$(C) $\left(\frac{\cot x}{1}+\frac{\cot ^{2} x}{2}+\ldots+\frac{\cot ^{9} x}{9}\right)$(D) $-\left(\frac{\cot x}{1}+\frac{\cot ^{2} x}{2}+\ldots+\frac{\cot ^{9} x}{9}\right)$
If $I n=\int \cos x \operatorname{cosec} x d x$, then $I n-I n_{-2}=$(A) $\frac{\cos (n-1) x}{n-1}$(B) $\frac{2 \cos (n-1) x}{n-1}$(C) $\frac{2 \sin (n-1) x}{n-1}$(D) none of these
If $I n=\int \frac{x^{n}}{\sqrt{x^{2}+a^{2}}} d x(n \geq 2)$, then$I n=\frac{x^{n-1} \sqrt{x^{2}+a^{2}}}{n}+k I n_{-2}$, where $k=$(A) $\frac{a^{2}(1-n)}{n} I_{n-2}$(B) $\frac{a^{2}(n-1)}{n} I_{n-2}$(C) $\frac{a^{2}(n+1)}{n} I_{n-2}$(D) none of these
If Im $n=\int \frac{\sin ^{\mathrm{m}} x}{\cos ^{\prime \prime} x} d x$, then$\operatorname{Im} n=\frac{\sin ^{\prime \prime \prime-1} x}{(n-1) \cos ^{n-1} x}+k \operatorname{lm}_{-2,} n_{-2}$, where $k=$(A) $\frac{m-1}{n-1}$(B) $\frac{1-m}{n-1}$(C) $\frac{m-1}{n}$(D) $\frac{m}{n-1}$
Column-I Column-II(A) $\int \sqrt{\sec x-1} d x$1. $\sin ^{-1}(\tan x)$(B) $\int \frac{d x}{\cos x \sqrt{\cos 2 x}}$2. $\sec ^{-1}(\sec x+\cos x)$(C) $\int \frac{d x}{\cos ^{3} x \sqrt{\sin 2 x}}$3. $-2 \log \left(\cos \frac{x}{2}+\sqrt{\cos ^{2} \frac{x}{2}-\frac{1}{2}}\right)$(D) $\int \frac{\sin ^{3} x d x}{\left(1+\cos ^{2} x\right) \sqrt{1+\cos ^{2} x+\cos ^{4} x}}$4. $\sqrt{2}\left(\sqrt{\tan x}+\frac{1}{5} \tan ^{5 / 2} x\right)$
Column-I Column-II(A) $\int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^{2}}}$1. $\frac{2(\sqrt{x}-1)}{\sqrt{1-x}}$(B) $\int\left(\frac{1-\sqrt{x}}{1+\sqrt{x}}\right)^{1 / 2} \frac{d x}{x}$2. $\frac{1}{\sqrt{2}} \sec ^{-1}\left(\frac{x^{2}+1}{\sqrt{2} x}\right)$(C) $\int x \sqrt{\frac{1+x}{1-x}} d x$3. $2 \cot ^{-1} \sqrt{x}-2 \ln \left|\frac{1+\sqrt{1-x}}{\sqrt{x}}\right|$(D) $\int \frac{\left(x^{2}-1\right)}{\left(x^{2}+1\right) \sqrt{x^{4}+1}}$4. $-\left(1+\frac{x}{2}\right) \sqrt{1-x^{2}}-\frac{1}{2} \cos ^{-1} x$
Column-I Column-II(A) $\int \frac{\sin ^{11} x-\cos ^{8} x}{1-2 \sin ^{2} x \cos ^{2} x} d x$1. $-\frac{2}{3} \sin ^{-1}\left(\cos ^{1 / 2} x\right)$(B) $\int \frac{\cos 5 x+\cos 4 x}{1-2 \cos 3 x} d x$2. $-\frac{1}{2} \sin 2 x$(C) $\int \sqrt{\frac{\cos x-\cos ^{3} x}{1-\cos ^{3} x}} d x$3. $\tan ^{-1}(\tan x-\cot x)$(D) $\int \frac{d x}{\sin ^{6} x+\cos ^{6} x}$4. $-\frac{\sin 2 x}{2}-\sin x$
Assertion:$\int e^{(x \sin x+\cos x)}\left(\frac{x^{2} \cos ^{2} x-x \sin x-\cos x}{x^{2}}\right) d x$$=e^{(x \sin x+\cos x)} \cdot \frac{\cos x}{x}+C$Reason: $\int e^{g(x)}\left\{f(x) g^{\prime}(x)+f^{\prime}(x)\right\} d x$$=e g^{\prime} x^{\prime} f(x)$
Assertion: $\int \frac{\sin ^{2} x}{a+b \cos x} d x=\frac{1}{b^{2}}(a x-b \sin x)$$$\begin{array}{l}-\frac{2 \sqrt{a^{2}-b^{2}}}{b^{2}} \tan ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right) \\ \text { Reason: } \int \frac{d x}{a+b \cos x} \\ \qquad \frac{2}{\sqrt{a^{2}-b^{2}}} \tan ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right)\end{array}$$
Assertion: $\int \sqrt{2+\tan ^{2} x} d x=\tan ^{-1}\left(\frac{\tan x}{\sqrt{2+\tan ^{2} x}}\right)$$+\frac{1}{2} \ln \left|\frac{\tan ^{2} x+\tan x+2}{\tan ^{2} x-\tan x+2}\right|$Reason: $\int \frac{2}{\left(1-x^{2}\right)\left(1+x^{2}\right)} d x$ $$=\tan ^{-1} x+\frac{1}{2} \ln \left|\frac{x+1}{x-1}\right|$$
Assertion: $\begin{aligned} \int x^{x+1} \log x(1+\log x) d x \\ &=x^{x}(x \log x-1)+C \end{aligned}$Reason: $\int x^{x}(1+\log x) d x=x^{x}$
$\int \frac{d x}{x\left(x^{n}+1\right)}$ is equal to:(A) $\frac{1}{n} \log \left(\frac{x^{\prime \prime}}{x^{n}+1}\right)+c$(B) $\frac{1}{n} \log \left(\frac{x^{n}+1}{x^{n}}\right)+c$(C) $\log \left(\frac{x^{n}}{x^{n}+1}\right)+c$(D) none of these
The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1+\alpha x)^{4}$ and of $(1-\alpha x)$ is the same if $\alpha$ equals $[2004]$(A) $-\frac{5}{3}$(B) $\frac{3}{5}$(C) $-\frac{3}{10}$(D) $\frac{10}{3}$
$\int\left\{\frac{(\log x-1)}{\left(1+(\log x)^{2}\right.}\right\} d x$ is equal to(A) $\frac{\log x}{(\log x)^{2}+1}+C$(B) $\frac{x}{x^{2}+1}+C$(C) $\frac{x e^{x}}{1+x^{2}}+C$(D) $\frac{x}{(\log x)^{2}+1}+C$
$\int \frac{d x}{\cos x+\sqrt{3} \sin x}$ equals(A) $\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+\mathrm{c}$(B) $\frac{1}{2} \log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+\mathrm{c}$(C) $\operatorname{logtan}\left(\frac{-}{2}-\frac{ }{12}\right)+\mathrm{c}$(D) $\log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+\mathrm{c}$
The value of $\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}$ is(A) $x+\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c \mathrm{x}$(B) $x-\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c$(C) $x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c$(D) $x-\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c$
If $\frac{d y}{d x}=y+3 ; y>-3$ and $y(0)=2$, then y $(\ln 2)$ is equal to(A) 5(B) 13(C) 2(D) 7
If the integral $\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k$then a is equal to(A) $-1$(B) $-2$(C) 1(D) 2
If $\int f(x) d x=\Psi(x)$ then $\int x^{5} f\left(x^{3}\right) d x$ is equal to(A) $\frac{1}{3} x^{3} \Psi\left(x^{3}\right)-3 \int x^{3} \Psi\left(x^{3}\right) d x+C$$[\mathbf{2 0 1 3}]$(B) $\frac{1}{3} x^{3} \Psi\left(x^{3}\right)-\int x^{2} \Psi\left(x^{3}\right) d x+C$(C) $\frac{1}{3}\left[x^{3} \Psi\left(x^{3}\right)-\int x^{3} \Psi\left(x^{3}\right) d x\right]+C$(D) $\frac{1}{3}\left[x^{3} \Psi\left(x^{3}\right)-\int x^{2} \Psi\left(x^{3}\right) d x\right]+C$
The integral $\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x$ is equal to $[2014]$(A) $(x-1) e^{x+\frac{1}{x}}+c$(B) $x e^{x+\frac{1}{x}}+c$(C) $(x+1) e^{x+\frac{1}{x}}+c$(D) $-x e^{x+\frac{1}{x}}+c$
The integral $\int \frac{d x}{x^{2}\left(x^{4}+1\right)^{3 / 4}}$ equals:[2015](A) $\left(x^{4}+1\right)^{1 / 4}+c$(B) $-\left(x^{4}+1\right)^{1 / 4}+c$(C) $-\left(\frac{x^{4}+1}{x^{4}}\right)^{1 / 4}+c$(D) $\left(\frac{x^{4}+1}{x^{4}}\right)^{1 / 4}+c$
The integral $\int \frac{2 x^{12}+5 x^{9}}{\left(x^{5}+x^{3}+1\right)}$ is equals to: $\quad$ [2016](A) $\frac{-x^{10}}{2\left(x^{5}+x^{3}+1\right)}+C$(B) $\frac{-x^{5}}{\left(x^{5}+x^{3}+1\right)^{2}}+C$(C) $\frac{-x^{10}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C$(D) $\frac{-x^{5}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C$