1.If \[I=\int\frac{dx}{\cos^{4}x-\cos^{2}x\sin^{2}x+\sin^{4}x}\]
then I is equals
a) \[2\tan^{-1}\left(\frac{\tan2x}{2}\right)+C\]
b) \[\tan^{-1}\left(\frac{\tan2x}{\sqrt{2}}\right)+C\]
c) \[\tan^{-1}\left(\frac{\tan2x}{2}\right)+C\]
d) \[\sqrt{2}\tan^{-1}\left(\frac{\tan2x}{\sqrt{2}}\right)+C\]
Explanation:
2. If \[I=\int\frac{dx}{\sqrt{\left(2 x-x^{2}\right)^{3}}}\]
then I is equals
a) \[\frac{2x-1}{\sqrt{x\left(2-x\right)}}+C\]
b) \[\frac{x-1}{\sqrt{x\left(2-x\right)}}+C\]
c) \[\frac{x+1}{\sqrt{x\left(2-x\right)}}+C\]
d) \[\frac{1-x}{\sqrt{x\left(2-x\right)}}+C\]
Explanation:
3.If \[I=\int\frac{dx}{\left(1+\sqrt{x}\right)\sqrt{x-x^{2}}}\]
then I is equals
a) \[\frac{2\left(\sqrt{x}+1\right)}{\sqrt{1-x}}+C\]
b) \[\frac{2\left(\sqrt{x}-1\right)}{\sqrt{1-x}}+C\]
c) \[\frac{2\left(1-\sqrt{x}\right)}{\sqrt{x\left(1-x\right)}}+C\]
d) \[\frac{2\left(\sqrt{x}-1\right)}{\sqrt{x\left(1-x\right)}}+C\]
Explanation:
4. If \[I=\int\left(\sqrt{\frac{a+x}{a-x}}+\sqrt{\frac{a-x}{a+x}}\right)dx\]
then I is equals
a) \[2\sin^{-1}\left(\frac{x}{a}\right)+C\]
b) \[2a\sin^{-1}\left(\frac{x}{a}\right)+C\]
c) \[2\cos^{-1}\left(\frac{x}{a}\right)+C\]
d) \[2a\cos^{-1}\left(\frac{x}{a}\right)+C\]
Explanation:
5. If \[I=\int\frac{dx}{x^{4}\sqrt{1+x^{2}}}\]
then I is equals
a) \[\frac{\sqrt{x^{2}+1}}{x}-\frac{1}{2x^{2}}+C\]
b) \[\frac{\sqrt{1+x^{2}}}{x}-\frac{1}{2x^{3}}+C\]
c) \[-\frac{\sqrt{1+x^{2}}}{x}+\frac{2x}{\sqrt{1+x^{2}}}+C\]
d) \[\frac{\sqrt{1+x^{2}}}{x}-\frac{2x}{\sqrt{1+x^{2}}}+C\]
Explanation:
6. If \[I=\int\frac{\sin\left( x+\alpha\right)+\cos x}{\sin\left(x-\alpha\right)}dx\]
then I is equals
a) \[2\cos\alpha\left(1+\sin\alpha\right)\log\mid\sin\left(x-\alpha\right)\mid+\left(\cos2\alpha+\cos\alpha\right)x+C\]
b) \[\cos\alpha\left(1+2\sin\alpha\right)\log\mid\sin\left(x-\alpha\right)\mid+\left(\cos2\alpha-\sin\alpha\right)x+C\]
c) \[2\sin\alpha\left(1+\cos\alpha\right)\log\mid\sin\left(x-\alpha\right)\mid+\left(\cos2\alpha+\cos\alpha\right)x+C\]
d) \[2\tan\alpha\left(1+\cos\alpha\right)\log\mid\sin\left(x-\alpha\right)\mid+\left(\cos\alpha-\sin\alpha\right)x+C\]
Explanation:
7. If \[I=\int e^{x}\frac{x^{3}+3x^{2}+4}{\left( x+1\right)^{3}}dx\]
then I is equals
a) \[ e^{x}\left(\frac{x^{2}-x+1}{\left( x+1\right)^{2}}\right)+C\]
b) \[ e^{x}\left(\frac{x^{2}-x+2}{\left( x+1\right)^{2}}\right)+C\]
c) \[ e^{x}\left(\frac{x^{2}+2x-2}{\left( x+1\right)^{2}}\right)+C\]
d) \[ e^{x}\left(\frac{x^{2}+2x-1}{\left( x+1\right)^{2}}\right)+C\]
Explanation:
8. If \[I=\int\tan x\tan2x\tan3x dx\]
then I is equals
a) \[\log \mid \cos x\mid +\left(1/2\right)\log \mid \cos 2x\mid+\left(1/3\right)\log\mid\cos 3x\mid+C\]
b) \[\log \mid \cos x\mid -\left(1/2\right)\log \mid \cos 2x\mid-\left(1/3\right)\log\mid\cos 3x\mid+C\]
c) \[\log \mid \cos x\mid +\left(1/2\right)\log \mid \cos 2x\mid-\left(1/3\right)\log\mid\cos 3x\mid+C\]
d) none of these
Explanation:
9. If \[I=\int\frac{dx}{\left(2\sin x+\sec x\right)^{4}}\]
then I is equals
a) \[-\frac{1}{5\tan^{5}x}+\frac{1}{3\tan^{6}x}-\frac{2}{7\tan^{7}x}+C\]
b) \[\frac{1}{5\tan^{5}x}+\frac{1}{3\tan^{6}x}-\frac{1}{\left(2\sin x+\sec x \right)^{3}}+C\]
c) \[\frac{-1}{3\left(2\sin x+\sec x\right)^{3}}+\tan^{-1}\left(3\sqrt{\tan x}\right)+C\]
d) \[\frac{-1}{3\left(2\sin x+\sec x \right)^{3}}-\tan^{-1}\left(3\sqrt{\tan x}\right)+C\]
Explanation:
10. If \[I=\int e^{x}\left(\frac{x+2}{x+4}\right)^{2}dx\]
then I is equals
a) \[ \frac{x}{x+4}e^{x}+C\]
b) \[ \frac{x-2}{x+4}e^{x}+C\]
c) \[ \frac{x+1}{x+4}e^{x}+C\]
d) \[ \frac{x-1}{x+4}e^{x}+C\]
Explanation:
