Evaluate the following integrals:

We know $d(\sin x)=\cos x$, and cot can be written in terms of $\cos$ and $\sin$

$\therefore \cot x=\frac{\cos x}{\sin x}$

$\therefore$ The given equation can be written as

$\Rightarrow \int \frac{\cos x}{\sin x \sqrt{\sin x}} d x$

$\Rightarrow \int \frac{\cos x}{\sin ^{3 / 2} x} d x$

Now assume $\sin x=t$

$d(\sin x)=d t$

$\cos x d x=d t$

Substitute values of $\mathrm{t}$ and $\mathrm{dt}$ in above equation

$\Rightarrow \int \frac{\mathrm{dt}}{\mathrm{t}^{2 \backslash 2}}$

$\Rightarrow \int \mathrm{t}^{-3 \backslash 2} \mathrm{dt}$

$\Rightarrow-2 t^{-1 \backslash 2}+c$

$\Rightarrow-2 \sin ^{-1 \backslash 2} x+c$

$\Rightarrow \frac{-2}{\sqrt{\sin x}}+c$