Integrate?

${\csc}^{2} \frac{x}{\cot} x \mathrm{dx}$

Feb 23, 2018

$\int {\csc}^{2} \frac{x}{\cot} x \mathrm{dx} = - \ln | \cot x | + \text{C}$

Explanation:

We can make a u-substitution:

Let $u = \cot \left(x\right) \implies \mathrm{du} = - {\csc}^{2} x \mathrm{dx}$

$\int {\csc}^{2} \frac{x}{\cot} x \mathrm{dx} \implies - \int \frac{1}{u} \mathrm{du}$

The integral of $\frac{1}{u}$ is simply $\ln | u | + \text{C}$

So $- \int \frac{1}{u} = - \ln | u | + \text{C}$

Undo the substitution:

$= - \ln | \cot x | + \text{C}$