Question

Integrate with respect to $x$:
$\frac{x}{x^2+1}$

Answer 1

Let $I=\int \frac{x}{x^2+1}dx$

Put $t=x^2+1$, then, $dt=2xdx$. So,

$I=\frac{1}{2}\int \frac{dt}{t}$

$=\frac{1}{2}\log \left|t\right|+c$

$=\frac{1}{2}\log \left|x^2+1\right|+c$