Question

Solve : $\int \frac{x^3}{(x^2+1{)}^3}dx$

Answer 1

$\int \frac{x^3}{{(x^2+1)}^3}dx$

Let, $t=x^2+1$

$dt=2xdx$

$\therefore$ $\frac{1}{2}\int \frac{x^2.2xdx}{{(x^2+1)}^3}$

$=\frac{1}{2}\int \frac{(t-1)dt}{t^3}$

$=\frac{1}{2}\int [\frac{1}{t^2}-\frac{1}{t^3}]dt$

$=\frac{1}{2}\int \frac{1}{t^2}dt-\frac{1}{2}\int \frac{1}{t^3}dt$

$=-\frac{1}{2t}+\frac{1}{{4t}^2}+C$

$=\frac{-1}{2(x^2+1)}+\frac{1}{{4(x^2+1)}^2}+C$