Mark the correct alternative in each of the following:
Question:

Mark the correct alternative in each of the following:

If $\int \frac{1}{(x+2)\left(x^{2}+1\right)} d x a \log \mid 1+x^{2}+b \tan ^{-1}$

$\mathrm{x}+\frac{1}{5} \log |\mathrm{x}+2|+\mathrm{C}$, then

A. $\mathrm{a}=-\frac{1}{10}, \mathrm{~b}=-\frac{2}{5}$

B. $a=\frac{1}{10}, b=-\frac{2}{5}$

C. $a=-\frac{1}{10}, b=\frac{2}{5}$

D. $a=\frac{1}{10}, b=\frac{2}{5}$

Solution:

$U=\int \frac{1}{(x+2)\left(x^{2}+1\right)} d x$

$U=\int \frac{A}{x+2} d x+\int \frac{B x+c}{x^{2}+1} d x$

$\frac{1}{(x+2)\left(x^{2}+1\right)}=\frac{A}{x+2}+\frac{B x+c}{x^{2}+1}$ (compare coefficient of $x^{2}$, and $x$ both side)

$\left[A=\frac{1}{5} ; B=-\frac{1}{5} ; C=\frac{2}{5}\right]$ put the value of $A, B, C$ in $U$

$U=\int \frac{\frac{1}{5}}{x+2} d x+\int \frac{-\frac{1}{5} x+\frac{2}{5}}{x^{2}+1} d x$

$U=\frac{1}{5}\left[\int \frac{1}{x+2} d x+\int \frac{-x}{x^{2}+1} d x+\int \frac{2}{x^{2}+1} d x\right]$

$U=\frac{1}{5}\left[\log (X+2)-\frac{1}{2} \log \left(x^{2}+1\right)+2 \tan ^{-1} X\right]+C$