Question

Find the antiderivative of $\frac{\cos \left(x\right)}{{\sin }^2\left(x\right)}$.

Answer 1

Compute the antiderivative:

We can find the antiderivative as

$\int \frac{\cos \left(x\right)}{{\sin }^2\left(x\right)}dx=\int \frac{1}{\sin \left(x\right)}·\frac{\cos \left(x\right)}{\sin \left(x\right)}dx$

$=\int cosec\left(x\right)·cot\left(x\right)dx[\because \frac{1}{\sin \left(x\right)}=cosec(x),\frac{\cos \left(x\right)}{\sin \left(x\right)}=cot(x)]$

$=-\cos ec\left(x\right)+C$ [ where $C$ is constant of integration]

Hence, antiderivative of $\frac{\cos \left(x\right)}{{\sin }^2\left(x\right)}$ is $-\cos ec\left(x\right)+C$, where $C$ is constant of integration.