1. Let \[I=\int\frac{x^{4}+4x^{3}+12x^{2}+9x}{\left(x+3\right)^{5}-x^{5}-243}dx\]
a) \[\frac{1}{15}\left[x-2 ln\left(x+3\right)\right]+c\]
b) \[\frac{1}{3}\left[x+2 ln\left(x+3\right)\right]+c\]
c) \[\frac{1}{15}\left[x+2 ln\left(x+3\right)\right]+c\]
d) \[\frac{1}{3}\left[2x- ln\left(x+3\right)\right]+c\]
Explanation:
2. Let \[I=\int\frac{x^{2}\sec^{2}x}{\left(x\tan x+1\right)^{2}}\]
then I is equal is
a) \[\frac{-x}{\cos x\left(x\sin x+\cos x\right)}+\tan x+c\]
b) \[\frac{-x\sec x}{x\sin x+\cos x}+\tan x\sec x+c\]
c) \[\frac{x\sec x}{x\sin x+\cos x}-\tan x+c\]
d) \[\frac{- x}{x\sin x+\cos x}-\tan x+c\]
Explanation:
3. Let \[I=\int\frac{dx}{\sin^{2} x+\tan^{2}x}\]
then
a) \[\cot x -\frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+c\]
b) \[-\frac{1}{2}\cot x -\frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+c\]
c) \[-\frac{1}{2}\cot x -\frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+c\]
d) \[-\cot x +\frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{\tan x}{2}\right)+c\]
Explanation:
4. Let \[I=\int\frac{dx}{x+\sqrt{2}\sqrt{x^{2}-1}}\]
then I is equal to
a) \[ln\left(x+\sqrt{x^{2}-1}\right)-\frac{1}{2}ln \mid\frac{x+\sqrt{2}\sqrt{x^{2}-1}}{x-\sqrt{2}\sqrt{x^{2}-1}}\mid+c\]
b) \[ln\left(x+\sqrt{x^{2}-1}\right)-\frac{1}{2\sqrt{2}}ln \mid\frac{x^{2}-2}{x-\sqrt{2}\sqrt{x^{2}-1}}\mid+c\]
c) \[\sqrt{2}ln\left(x+\sqrt{x^{2}-1}\right)-ln \mid x+\sqrt{2}\sqrt{x^{2}-1}\mid+c\]
d) \[\sqrt{2}ln\left(x+\sqrt{x^{2}-1}\right)+ln \mid\frac{\left(2-x^{2}\right)\left(x+\sqrt{2}\sqrt{x^{2}-1}\right)}{x-\sqrt{2}\sqrt{x^{2}-1}}\mid+c\]
Explanation:
5. Let \[I=\int\left(x^{10}+\sqrt{1+x^{20}}\right)^{\frac{21}{10}}dx\] and
\[u=\sqrt{x^{-20}+1}\]
then I is equal to
a) \[u^{\frac{1}{20}}\left(1+u\right)-\frac{2}{9}\left(1+u\right)^{\frac{19}{20}}+c\]
b) \[2u\left(1+u\right)^{-\frac{19}{20}}-\frac{4}{9}u^{-\frac{1}{20}}+c\]
c) \[u\left(1+u\right)^{-\frac{1}{20}}-\frac{4}{9}\left(1+u\right)^{\frac{19}{20}}+c\]
d) \[2u\left(1+u\right)^{-\frac{1}{20}}-\frac{40}{9}\left(1+u\right)^{\frac{19}{20}}+c\]
Explanation:
6. \[\int\frac{\sin x}{\sin 4x}dx=A\log\mid \frac{1+\sin x}{1-\sin x}\mid +B\log\mid \frac{1+\sqrt{2}\sin x}{1-\sqrt{2}\sin x}\mid +c\]
a) \[A=\frac{1}{8},B=\frac{1}{4\sqrt{2}}\]
b) \[A=-\frac{1}{8},B=-\frac{1}{4\sqrt{2}}\]
c) \[A=-\frac{1}{8},B=\frac{1}{4\sqrt{2}}\]
d) \[A=\frac{1}{8},B=-\frac{1}{4\sqrt{2}}\]
Explanation:
7. Let \[I=\int\left(\sqrt{\cot x}-\sqrt{\tan x}\right) dx\]
then I is equal to
a) \[\sqrt{2}\log\left(\sqrt{\tan x}-\sqrt{\cot x}\right) +C\]
b) \[\sqrt{2}\log\mid\sin x+\cos x+\sqrt{\sin 2x} \mid +C\]
c) \[\sqrt{2}\log\mid\sin x-\cos x+\sqrt{ 2}\sin x\cos x \mid +C\]
d) \[\sqrt{2}\log\mid\sin \left(x+\pi/4\right)+\sqrt{ 2}\sin x\cos x \mid +C\]
Explanation:
8. If \[I=\int\sqrt{\tan x} dx\]
then I is equal to
a) \[\frac{1}{\sqrt{2}}\left[\sin^{-1}\left(\sin x-\cos x\right)+\log \mid\sin x+\cos x+\sqrt{\sin 2x}\mid\right]+C\]
b) \[\sqrt{2}\left[\sin^{-1}\left(\sin x+\cos x\right)+\log \mid\sin x-\cos x+\sqrt{\sin 2x}\mid\right]+C\]
c) \[\sqrt{2}\left[\sin^{-1}\left(\cos x-\sin x\right)+\log \sin 2x\right]+C\]
d) \[\sin^{-1}\left(\cos x-\sin x\right)+2\log\sin 2x+C\]
Explanation:
9. If \[I=\int\frac{dx}{x\sqrt{1-x^{3}}}\]
then I is equal to
a) \[\frac{1}{3}\log \mid\frac{\sqrt{1-x^{3}}-1}{\sqrt{1-x^{3}}+1}\mid+C\]
b) \[\frac{1}{3}\log\mid \frac{\sqrt{1-x^{3}}+1}{\sqrt{1-x^{3}}-1}\mid+C\]
c) \[\frac{2}{3}\log \mid1-x^{3} \mid+C\]
d) \[\frac{1}{3}\log \mid x^{3/2}+\sqrt{1-x^{3}} \mid+C\]
Explanation:
10. If \[I=\int\frac{x^{5}}{\sqrt{1+x^{3}}}dx\]
then I is equal to
a) \[\frac{2}{9}\left(1+x^{3}\right)^{5/2}+\frac{2}{3}\left(1+x^{3}\right)^{3/2}+C\]
b) \[\frac{2}{9}\left(1+x^{3}\right)^{3/2}-\frac{2}{3}\left(1+x^{3}\right)^{1/2}+C\]
c) \[\log\mid\sqrt{x}+\sqrt{1+x^{3}}\mid +C\]
d) \[x^{2} \log \left(1+x^{3}\right)+C\]
Explanation:
