Question

\[\int\frac{\cot x}{\sqrt{\sin x}} dx\]

Answer 1

\[\int\frac{\cot x}{\sqrt{\sin x}}dx\]

\[ = \int\frac{\cos x}{\sin x \sqrt{\sin x}} dx\]

\[ = \int\frac{\cos x}{\left( \sin x \right)^\frac{3}{2}}dx\]

\[Let \sin x = t\]

\[ \Rightarrow \cos x = \frac{dt}{dx}\]

\[ \Rightarrow \text{cos x dx} = dt\]

\[Now, \int\frac{\cos x}{\left( \sin x \right)^\frac{3}{2}}dx\]

\[ = \int\frac{dt}{t^\frac{3}{2}}\]

\[ = \int t^{- \frac{3}{2}} dt\]

\[ = \left[ \frac{t^{- \frac{3}{2} + 1}}{\frac{- 3}{2} + 1} \right] + C\]

\[ = \frac{- 2}{\sqrt{t}} + C\]

\[ = - \frac{2}{\sqrt{\sin x}} + C\]