1. If \[I=\int\frac{dx}{\left( 1+x^{4}\right)^{1/4}}\]
then I is equals
a) \[\frac{1}{2}\tan^{-1}\left(\frac{x}{\left(1+x^{4}\right)^{1/4}}\right)+C\]
b) \[\frac{1}{4}\log\mid\frac{1-\left(1+x^{4}\right)^{1/4}}{1+\left(1+x^{4}\right)^{1/4}}\mid+C\]
c) \[\frac{1}{2}\tan^{-1}\left(\frac{\left(1+x^{4}\right)^{1/4}}{x}\right)-\frac{1}{4}\log\left(\frac{1-\left(1+x^{4}\right)^{1/4}}{1+\left(1+x^{4}\right)^{1/4}}\right)+C\]
d) \[\frac{1}{2}\tan^{-1}\left(\frac{\left(1+x^{4}\right)^{1/4}}{x}\right)-\frac{1}{4}\log\left(\frac{x-\left(1+x^{4}\right)^{1/4}}{x+\left(1+x^{4}\right)^{1/4}}\right)+C\]
Explanation:
2. If \[I=\int\tan^{-1}\left(1+\sqrt{x}\right)dx\]
then I is equals
a) \[x ^{2}\tan^{-1}\left(1+\sqrt{x}\right)-\sqrt{x}+\log\left(x+2\sqrt{x}+2\right)+C\]
b) \[\left(2\sqrt{2}+1\right)\tan^{-1}\left(1+\sqrt{x}\right)+\sqrt{x}-\log\left(x+2\sqrt{x}+2\right)+C\]
c) \[\left(\sqrt{x-1}\right)^{2}\tan^{-1}\left(1+\sqrt{x}\right)-\sqrt{x}+\log\left(\sqrt{x}+1\right)+C\]
d) none of these
Explanation:
3. If \[I=\int\frac{2x+3}{\left(x^{2}+2x+3\right)\sqrt{x^{2}+2x+4}}dx\]
then I is equals
a) \[ \log\mid\frac{\sqrt{x^{2}+2x+4}-1}{\sqrt{x^{2}+2x+4}+1}\mid+C\]
b) \[ \log\mid\frac{\sqrt{x^{2}+2x+4}-1}{\sqrt{x^{2}+2x+4}+1}\mid+\tan^{-1}\left(\frac{x+2}{3}\right)+C\]
c) \[ \log\tan^{-1}\left(\frac{\sqrt{x+3}}{2}\right)+C\]
d) none of these
Explanation:
4. If \[I=\int\cot^{-1}\left(\frac{a^{2}-ax+x^{2}}{a^{2}}\right)dx\]
then I is equals
a) \[x \tan^{-1}\left(\frac{x}{a}\right)-\left(x-a\right)\tan^{-1}\left(\frac{x-a}{a}\right)+C\]
b) \[\frac{a}{2}\log\left(2a^{2}-2ax+x^{2}\right)-\frac{a}{2}\log\left(x^{2}+a^{2}\right)+C\]
c) \[x \tan^{-1}\left(\frac{x}{a}\right)+\left(x-a\right)\tan^{-1}\left(\frac{x-a}{a}\right)+\frac{a}{2}\log\left(2a^{2}-2ax+x^{2}\right)+C\]
d) \[\tan^{-1}\left(\frac{x}{a}\right)+x\tan^{-1}\frac{x-a}{a}+\log\left(2a^{2}-2ax+x^{2}\right)+C\]
Explanation:
5. If \[I=\int\frac{\cos x}{\sin ^{3}x-\cos^{3} x}dx\]
and \[f\left(x\right)=\frac{1}{3}\log\left(\tan x-1\right)+\frac{1}{\sqrt{3}}\tan^{-1}\left(\frac{2\tan x+1}{\sqrt{3}}\right),g\left(x\right)=\frac{1}{6}\log\left(\tan^{2}x+\tan x+1\right)\]
then I is equals
a) f(x) – g(x) + C
b) f(x) g(x) + C
c) f(x)/g(x) + C
d) f(x) + g(x) + C
Explanation:
6. If \[I=\int\frac{\sqrt{\sin^{3}2x}}{\sin^{5}x}dx\]
and \[f\left(x\right)=\left(\cot x\right)^{3/2},g\left(x\right)=\left(\cot x\right)^{5/2}\]
then I is equals
a) \[\left(2\sqrt{3}/3\right) f\left(x\right) -\left(1/5\right) g\left(x\right)+C\]
b) \[-\left(4\sqrt{2}/5\right)g\left(x\right) +C\]
c) \[\left(1/2\sqrt{3}\right)f\left(x\right) +C\]
d) \[\left(2\sqrt{2}/3\right)f\left(x\right)+ 1/5f\left(x\right)+C\]
Explanation:
7. If \[\int\frac{dx}{\left(x^{2}+1\right)\left(x^{2}+4\right)}=K\tan^{-1}x+L\tan^{-1}\frac{x}{2}+C\]
then
a) K = 1/3
b) L = 2/3
c) L = – 1/6
d) Both a and c
Explanation:
8. If \[\int\frac{\cos^{4}x}{\sin^{2}x}dx=K\cot x+M\sin 2x+L\frac{x}{2}+C\]
then
a) L = – 3
b) K = – 2
c) M = – 1/4
d) Both a and c
Explanation:
9. If \[\int\frac{3x+4}{x^{3}-2x-4}dx=\log\mid x-2\mid+K\log f\left(x\right)+C\]
then
a) K = – 1/2
b) \[f\left(x\right)= x^{2}+2x+2 \]
c) \[f\left(x\right)=\mid x^{2}+2x+2 \mid\]
d) All of the Above
Explanation:
10. If \[\int\frac{dx}{5+4\cos x}=K\tan^{-1}\left(M\tan\frac{x}{2}\right)+C\]
then
a) K = 1
b) K = 2/3
c) M = 1/3
d) Both b and c
Explanation:
