Question

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

Answer 1

Consider the following question.

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

Let, and differentiate w.r.t to dx we get.

$t=(4x-1)$

$\frac{dt}{dx}=4$

$=\frac{1}{4}\underset{}{\overset{}{\int }}\frac{1}{t\sqrt{t^2-1}}dt$

$=\frac{1}{4}{\sec }^{-1}\left(t\right)$

$=\frac{1}{4}{\sec }^{-1}(4x-1)+C$

Hence, this is the required answer.