PrepBharat

1.If \[I=\int\frac{\left(x+1\right)^{2}}{\sqrt{x^{2}+1}}dx\]
then 2I is equal to
a) \[\left(x+4\right)\sqrt{x^{2}+1}+\log\left(x+\sqrt{x^{2}+1}\right)+C\]
b) \[x\sqrt{x^{2}+1}+2\log\left(x+\sqrt{x^{2}+1}\right)+C\]
c) \[x\sqrt{x^{2}+1}+\log\left(x+\sqrt{x^{2}+1}\right)+C\]
d) \[\left(x-3\right)\sqrt{x^{2}+1}+\log\left(x+\sqrt{x^{2}+1}\right)+C\]

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2. If \[I=\int\frac{x^{2}}{\left(x-a\right)\left(x-b\right)}dx\]
then I is equals
a) \[x+\frac{1}{a-b}\log \mid \frac{x-a}{x-b}\mid+C\]
b) \[x+\frac{1}{a-b}\log \mid \frac{x-a}{x-b}\mid^{a^{2}+b^{2}}+C\]
c) \[x+\frac{1}{a-b}\left\{a^{2}\log \mid x-a\mid -b^{2}\log \mid x-b\mid\right\}+C\]
d) \[x^{2}+\frac{1}{a-b}\log \mid \frac{x-a}{x-b}\mid+C\]

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3.If \[I=\int\cos\theta\log\left(\tan\theta/2\right)d\theta\]
then I is equal to
a) \[\sin\theta\log\left(\tan\theta/2\right)+\theta+C\]
b) \[\cos\theta\log\left(\tan\theta/2\right)+\theta+C\]
c) \[\sin\theta\log\left(\tan\theta/2\right)-\theta+C\]
d) \[\cos\theta\log\left(\tan\theta/2\right)-\theta+C\]

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4. If \[I=\int\frac{dx}{x\sqrt{x^{4}+1}}\]
then I is equals
a) \[\left(1/4\right)\log\mid \frac{\sqrt{x^{4}+1}-1}{\sqrt{x^{4}+1}+1}\mid+C\]
b) \[\left(1/4\right)\log\mid \frac{\sqrt{x^{4}+1}+1}{\sqrt{x^{4}+1}-1}\mid+C\]
c) \[\left(1/4\right) \left(x^{4}+1\right)^{3/2}+C\]
d) \[\left(1/4\right) \left(x^{4}-1\right)^{-3/2}+C\]

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5. If \[I=\int\frac{dx}{\tan x\log cosec x}\]
then I is equals
a) log |log cosec x| + C
b) log |log cos x| + C
c) – log |log (cosec x)| + C
d) log |log sin x| + C

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6. If \[I=\int\frac{\sin x-\cos x}{\sqrt{\sin 2x}}dx\]
then I is equals
a) \[\log \mid \sin x-\cos x+\sqrt{\sin 2x}\mid +C\]
b) \[\log \mid \sin x+\cos x-\sqrt{\sin 2x}\mid +C\]
c) \[\log \mid \sin x+\cos x\mid +C\]
d) \[\log \mid \sin x-\cos x\mid +C\]

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7. If \[I=\int\frac{\cos x dx}{\sqrt{a+b \cot^{2}x}}\left(a>b>0\right)\]
then I is equals
a) \[\frac{1}{a-b}\sqrt{a \sin^{2}x+ b \cos^{2}x}+C\]
b) \[\frac{1}{a-b}\sqrt{a + b\cot^{2}x}+C\]
c) \[\frac{1}{a-b}\left(\sqrt{a + b\cot^{2}x}+x\right)+C\]
d) \[\frac{1}{a-b}\left(\sqrt{a + b\cot^{2}x}-x\right)+C\]

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8. If \[I=\int\frac{dx}{\left(e^{x}+2\right)^{3}}\]
then I is equals
a) \[\frac{1}{8}x-\frac{1}{8}\log\left(e^{x}+2\right)+\frac{e^{x}+3}{4\left(e^{x}+2\right)^{2}}+C\]
b) \[\frac{1}{8}x+\frac{1}{8}\log\left(e^{x}+2\right)+\frac{e^{x}}{4\left(e^{x}+2\right)^{2}}+C\]
c) \[\frac{1}{8}x+\frac{1}{8}\log\left(e^{x}+2\right)+\frac{e^{x}}{\left(e^{x}+2\right)^{2}}+C\]
d) none of these

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9. If \[I=\int\frac{\tan x}{\tan3 x}dx\]
then I is equals
a) \[x-\frac{1}{\sqrt{3}}\log\mid\frac{\sqrt{3}+\tan x}{\sqrt{3}-\tan x}\mid+C\]
b) \[x+\frac{1}{\sqrt{3}}\log\mid\frac{\sqrt{3}+\tan x}{\sqrt{3}-\tan x}\mid+C\]
c) \[x-\frac{2}{\sqrt{3}}\log\mid\frac{3-\tan x}{3+\tan x}\mid+C\]
d) \[x+\frac{2}{\sqrt{3}}\log\mid\frac{3-\tan x}{3+\tan x}\mid+C\]

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10. If \[I=\int\frac{x^{2}+a^{2}}{x^{4}-a^{2}x^{2}+a^{4}}dx\]
a) \[\frac{1}{a}\tan^{-1}\left(\frac{ax}{x^{2}-a^{2}}\right)+C\]
b) \[\frac{1}{a}\tan^{-1}\left(\frac{x^{2}-a^{2}}{ax}\right)+C\]
c) \[\log\mid x+\sqrt{x^{2}-a^{2}}\mid+x+C\]
d) \[\log\mid x+\sqrt{x^{2}+a^{2}}\mid+\tan^{-1}\frac{ax}{x^{2}-a^{2}}+C\]

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