Question

$\int \frac{\cot \mathrm{x}}{\sqrt{\sin x}}dx$

Answer 1

$$\begin{gathered} \int \frac{\cot x}{\sqrt{\sin x}}dx \\ =\int \frac{\cos x}{\sin x\sqrt{\sin x}}dx \\ =\int \frac{\cos x}{{\left(\sin x\right)}^{{\displaystyle \frac{3}{2}}}}dx \\ \mathrm{Let}\sin x=t \\ \Rightarrow \cos x=\frac{dt}{dx} \\ \Rightarrow \cos xdx=dt \\ \mathrm{Now},\int \frac{\cos x}{{\left(\sin x\right)}^{{\displaystyle \frac{3}{2}}}}dx \\ =\int \frac{dt}{t^{{\displaystyle \frac{3}{2}}}} \\ =\int t^{-\frac{3}{2}}dt \\ =\left[\frac{t^{-{\displaystyle \frac{3}{2}}+1}}{{\displaystyle \frac{-3}{2}}+1}\right]+\mathrm{C} \\ =\frac{-2}{\sqrt{t}}+\mathrm{C} \\ =-\frac{2}{\sqrt{\sin x}}+\mathrm{C} \end{gathered}$$