Question

Evaluate: $\int \frac{dt}{\sqrt{(t+\alpha )(t+\beta )}}$

Answer 1

Put $t+\alpha =z$, we get

$\int \frac{dz}{\sqrt{z(z-n)}}\text{where}n=\alpha -\beta$

Put $z=n{\sec }^2\theta$

$\Rightarrow dz=2n{\sec }^2\theta \tan \theta$

$\Rightarrow \int \frac{2n{\sec }^2\theta \tan \theta }{\sqrt{n^2{\sec }^2\theta {\tan }^2\theta }}d\theta$

$=\int 2\sec \theta d\theta$ where $\theta ={\sec }^{-1}\sqrt{\frac{t+\alpha }{\alpha -\beta }}$