PrepBharat

1.Let a, b > 0. For x > 0, let \[f\left(x\right)=ln\left[\frac{\left(x+a\right)^{x+a}}{\left(x+b\right)^{x+b}}\right]\]     and g(x) = (x + a) (x + b) ln (x + a) ln (x + b), then \[\int\frac{f\left(x\right)}{g\left(x\right)} dx\]    is equal to
a) \[\frac{ln\left(x+b\right)}{ln\left(x+a\right)} +C\]
b) \[ln\left(\frac{ln\left(x+b\right)}{ln\left(x+a\right)}\right) +C\]
c) ln (ln (x + b) – ln (x + a)) + C
d) \[ln\left(\frac{\left(x+a\right)}{\left(x+b\right)}\right)-\frac{ln\left(x+b\right)}{ln\left(x+a\right)} +C\]

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2. Let \[I=\int\frac{\left(\sqrt{1+x+x^{2}}-1\right)^{2}}{x^{2}\sqrt{1+x+x^{2}}}dx\]
then
a) \[\frac{2}{x}\left(1-\sqrt{1+x+x^{2}}\right)+C\]
b) \[\frac{2}{x}\left(1-\sqrt{1+x+x^{2}}\right)+ln\left(x+\frac{1}{2}+\sqrt{x^{2}+x+1}\right)+C\]
c) \[\frac{-2\left(1+x\right)}{\sqrt{1+x+x^{2}+1}}+ln\left(x+\sqrt{x^{2}+x+1}\right)+C\]
d) \[-\frac{1}{x}\left(1+\frac{1}{\sqrt{1+x+x^{2}}}\right)+ln\left(x+\sqrt{x^{2}+x+1}\right)+C\]

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3.If \[I=\int\frac{1}{x^{1/2}-x^{1/4}}+\frac{\log\left(1+x^{1/4}\right)}{x^{1/2}+x^{1/4}}dx\]
then I is equal to
a) \[2\sqrt{x}+4x^{1/4}+\log \mid x^{1/4}-1\mid+C\]
b) \[2\left(x^{1/2}-6x^{1/4}+3\right)\log ( x^{1/4}+1)+C-\left(x^{1/2}-10x^{1/4}+7\right)+2\left\{\log\left(x^{1/4}+1\right)\right\}^{2}\]
c) \[\sqrt{x}-x^{1/4}+6\log\left(x^{1/4}+1\right) -3x^{3/4}+C\]
d) none of these

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4. If \[I=\int\sqrt{\frac{\cos^{3}x}{\sin^{11}x}}dx\]
then I is equal to
a) \[-\frac{2}{5}\left(\cot x\right)^{5/2}-\frac{2}{7}\left(\cot x\right)^{7/2}+C\]
b) \[-\frac{7}{5}\left(\cot x\right)^{7/2}-\frac{3}{11}\left(\cot x\right)^{11/3}+C\]
c) \[\frac{5}{2}\left(\cot x\right)^{5/2}+\frac{11}{2}\left(\cot x\right)^{2/11}+C\]
d) none of these

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5. If \[I=\int\left[1+\cot\left(x-\alpha\right)\cot\left(x+\alpha\right)\right]dx\]
then I is equal to
a) \[\log \mid\frac{\cot x-\cot\alpha}{\cot x+\cot\alpha} \mid+C\]
b) \[\cot 2\alpha\log \mid\frac{1-\cot x\tan\alpha}{1+\cot x\tan\alpha} \mid+C\]
c) cosec \[2\alpha\log \mid\frac{\tan x-\cot\alpha}{\tan x+\cot\alpha} \mid+C\]
d) \[\log \mid\tan x \mid-x \log\mid\tan\alpha \mid+C\]

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

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

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8. If \[I=\int\frac{\log\left(\cos x\right)}{\cos^{2}x}\]
then I is equal
a) tan x log cos x + tan x – x + C
b) \[\tan x \log \cos x - \tan x + x^{2} + C\]
c) tan x log cos x – cot x + x + C
d) tan x log cos x + cot x – x + C

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

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

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