Evaluate the following integral:

Question:

Evaluate the following integral:

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

Solution:

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

$\frac{\mathrm{x}}{(\mathrm{x}+1)\left(\mathrm{x}^{2}+1\right)}=\frac{\mathrm{A}}{\mathrm{x}+1}+\frac{\mathrm{Bx}+\mathrm{C}}{\mathrm{x}^{2}+1}$

$x=A\left(x^{2}+1\right)+(B x+C)(x+1)$

Equating constants

$0=A+C$

Equating coefficients of $x$

$1=B+C$

Equating coefficients of $x^{2}$

$0=A+B$

Solving, we get

$\mathrm{A}=-\frac{1}{2} \mathrm{~B}=\frac{1}{2} \mathrm{C}=\frac{1}{2}$

Thus

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

$I=-\frac{1}{2} \log |x+1|+\frac{1}{4} \log \left|x^{2}+1\right|+\frac{1}{2} \tan ^{-1} x+C$

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