Question

Evaluate

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

Answer 1

We have,

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

So,

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

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

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

On integrating and we get,

$\frac{{(x^2+1)}^{\frac{-3}{2}}}{\frac{-3}{2}}+C$

$\Rightarrow -\frac{2{(x^2+1)}^{\frac{-3}{2}}}{3}+C$

Hence, this is the answer.