Question

$\int \frac{1}{\sqrt{{\sin }^{3}x\sin (x+a)}}dx$

Answer 1

$\int \frac{1}{\sqrt{si{n}^{3}x[\sin x\cos \alpha +\cos x\sin x]}}dx$

$\int \frac{1}{\sqrt{{\sin }^4[\cos \alpha +\cot x\sin \alpha ]}}dx$

$\int \frac{cose{c}^{2}xdx}{\sqrt{\cos \alpha +\sin \alpha \cot x}}dx$

let $\cot x=t$

$-cose{c}^{2}xdy=dt$

$\int \frac{-dt}{\sqrt{{\cos }^{2}x+\sin \alpha }}dt$

$=-\frac{2\sqrt{\cos \alpha +t\sin \alpha }}{\sin \alpha }+c$

$=\frac{-2\sqrt{\cos \alpha +\cot x\sin x}}{\sin \alpha }+c$