Question

Evaluate $\int \frac{1}{(4x-1)\sqrt{{(4x-1)}^2-1}}dx$

Answer 1

$\int \frac{1}{(4x-1)\sqrt{(4x-1{)}^2-1}}dx................\left(1\right)$

put $t=4x-1⟹dt=4dx$

$\left(1\right)$ becomes $\int \frac{1}{t\sqrt{t^2-1}}\frac{dt}{4}=\frac{1}{4}{\text{sec}}^{-1}t=\frac{1}{4}{\text{sec}}^{-1}(4x-1)$