Question

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

Answer 1

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

$x^2-1=t$ $2xdx=dt$ $xdx=\frac{1}{2}dt$

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

$\sqrt{t}-ln(x+\sqrt{x^2-1})+c$

$\sqrt{x^2-1}-ln(x+\sqrt{x^2-1})+c$