Question

Solve
$\int \sqrt{\frac{a-x}{x-b}}dx$

Answer 1

$\begin{array}{c}\int \sqrt{\frac{a-x}{x-b}}dxt=\frac{a-x}{x-b}\Rightarrow x=\frac{a+bt}{t+1}dx=-\frac{\left(a-b\right)}{{\left(t+1\right)}^2}dt\\ \Rightarrow -\int \sqrt{t}.\frac{\left(a-b\right)}{{\left(t+1\right)}^2}dt=-\left(a-b\right)\int \frac{\sqrt{t}}{{\left(t+1\right)}^2}dt\\ t={\tan }^{2}u\Rightarrow dt=2\tan u{\sec }^{2}udu\\ I=-\left(a-b\right)\int \frac{\tan u}{\left({\tan }^{2}u+1\right)}2.2\tan u{\sec }^{2}udu\\ =-2\left(a-b\right)\int {\sin }^{2}udu\\ =-2\left(a-b\right)\int \left(\frac{1-{\cos }^{2}u}{2}\right)du\\ \Rightarrow -\left(a-b\right)\left[u-\frac{\sin 2u}{2}\right]+c\end{array}$