Question

Evaluate ${\int }^{}\frac{dx}{(2x-7)\sqrt{(x-3)(x-4)}}$

Answer 1

$\int \frac{dx}{(2x-7)\sqrt{\left(x-3\right)\left(x-4\right)}}$

$\int \frac{dx}{\left(2x-7\right)\sqrt{\left(x-3\right)(x-4)}}$

$\int \frac{dx}{(2x-7)\sqrt{x^2-7x+12}}$

$u=\sqrt{x^2-7x+12}$

$dx=\frac{2\sqrt{x^2-7x+12}}{(2x-7)}dx$

$2\int \frac{du}{4u^2+1}$

$2{\tan }^{-1}\left(2u\right)$

$2{\tan }^{-1}\left(2\sqrt{x^2-7x+12}\right)+C$