Integrate

cot(x) / {(sinx)^1/2}

Let I = \int \frac{\cot x}{\sqrt{\sin x}} = \int \frac{\cos x}{{(\sin x)}^{3 / 2}} dx Put \sin x = t \Rightarrow \cos x dx = dt So, I = \int \frac{dt}{t^{3 / 2}} = \int t^{- \frac{3}{2}} dt = \frac{t^{- \frac{3}{2} + 1}}{1 - \frac{3}{2}} + C = - \frac{1}{2 \sqrt{t}} + C = - \frac{1}{2 \sqrt{\sin x}} + C

Integrate cot(x) / {(sinx)1/2}