Question

Solve:

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

Answer 1

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

Put $x=1/t$

$\Rightarrow dx=\frac{-1}{t^2}dt$

$\int \frac{-dt}{t^2(\frac{1}{t^2}+1)\sqrt{\frac{1}{t^2}-1}}$

$I=\int \frac{-tdt}{(1+t^2)(1-t^2)}$

let $1-t^2=z^2$

$\Rightarrow -2tdt=2zdz$

$\Rightarrow I=\int \frac{zdz}{(2-z^2)z}$

$=\int \frac{dz}{2-z^2}$

$=\frac{1}{2\sqrt{2}}ln\left|\frac{\sqrt{2}+z}{\sqrt{2}-z}\right|+c_1$

$=\frac{1}{2\sqrt{2}}ln\left|\frac{\sqrt{2}+\sqrt{x^2-1}}{\sqrt{2}-\sqrt{x^2-1}}\right|+c_1$