Evaluate the following integral:
$\int \frac{3}{(1-x)\left(1+x^{2}\right)} d x$
$I=\int \frac{3}{(1-x)\left(1+x^{2}\right)} d x$
$\frac{3}{(1-x)\left(1+x^{2}\right)}=\frac{A}{1-x}+\frac{B x+C}{1+x^{2}}$
$3=A\left(1+x^{2}\right)+(B x+C)(1-x)$
Equating similar terms
$A-B=0$
$B-C=0$
$A+C=3$
Solving
$A=\frac{3}{2}, B=\frac{3}{2}, C=\frac{3}{2}$
Thus,
$I=\frac{3}{2} \int \frac{d x}{1-x}+\frac{3}{2} \int \frac{x d x}{1+x^{2}}+\frac{3}{2} \int \frac{d x}{1+x^{2}}$
$=-\frac{3}{2} \log |1-x|+\frac{3}{2} \log \left|1+x^{2}\right|+\frac{3}{2} \tan ^{-1} x+C$
$I=\frac{3}{4}\left[\log \left|\frac{1+x^{2}}{(1-x)^{2}}\right|+2 \tan ^{-1} x\right]+C$
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