Question

Integrate the function: $\frac{1}{\sqrt{sin^{3}xsin(x+α)}}$

Answer 1

\frac{1}{\sqrt{sin^{3}xsin(x+α)}}$$\frac{1}{\sqrt{sin^{3}x(sinxcosα+cosxsinα)}}

$=\frac{1}{\sqrt{sin^{4}xcosα+sin^{3}x cosx sinα)}}$

$=\frac{1}{sin^{2}x\sqrt{cosα+cotx sinα}}$

$=\frac{cosec^{2}x}{\sqrt{cosα+cotx sinα}}$

Let cosα+cotx sinα=t$$⇒-cosec^{2}x sinα dx=dt

$∴∫\frac{1}{sin^{3}xsin(x+α)}dx$=$∫\frac{cosec^{2}x}{\sqrt{cosα+cotx sinα}} dx$

$=\frac{-1}{sinα∫}\frac{dt}{\sqrt{t}}$

$=\frac{-1}{sinα}[2\sqrt{t}]+C$

$=\frac{-1}{sinα}[2\sqrt{cosα+cotx sinα}]+C$

$=\frac{-2}{sinα}√\sqrt{cosα+\frac{cosx sinα}{sinx}}+C$

$=\frac{-2}{sinα}\sqrt{\frac{sinx cosα+cosx sinα}{sinx}}+C$

$=\frac{-2}{sinα}\sqrt{\frac{sin(x+α)}{sinx}}+C$