Question

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

Answer 1

Given ,$\int \frac{1}{(2x-7)\sqrt{(x-3)(x-4)}}dx$

Let,

$y=\int \frac{1}{(2x-7)\sqrt{(x-3)(x-4)}}dx$

$y=\int \frac{1}{(2x-3)\sqrt{x^2-7x+12}}dx$

Put

$2x-7=\frac{1}{t}\Rightarrow x=\frac{1+7t}{2t}$

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

$dx=-\frac{1}{2t^2}$

Now.

$y=\int \frac{1}{\frac{1}{t}\sqrt{{\left(\frac{1+7t}{2t}\right)}^2-7\left(\frac{1+7t}{2t}\right)+12}}\times (-\frac{1}{2t^2})dt$

$y=\int \frac{2t^2}{\sqrt{{(1+7t)}^2-7(1+7t)\times 2t+12\times 4t^2}}\left(\frac{1}{-2t^2}\right)dt$

$y=-\int \frac{1}{\sqrt{1-t^2}}dt$

$\because \int \frac{1}{\sqrt{a^2-x^2}}dx={\sin }^{-1}\frac{x}{a}+c$

$\therefore y=-{\sin }^{-1}t+c$

$y=-{\sin }^{-1}\frac{1}{2x-7}+c$