Question

$\int \frac{\log \left({\sec }^{-1}x\right)}{x\sqrt{x^2-1}}dx$

Answer 1

$\int \frac{log\left({sec}^{-1}x\right)}{x\sqrt{x^2-1}}dx$

$\to$ Putting ${sec}^{-1}x=t$

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

$\to \int logtdt$

$v=1;u=logt$

$\int v=t;\frac{du}{dt}=\frac{1}{t}$

$=tlogt-\int t+c$

$=tlogt-t+c$