Question

Evaluate: $\int x\cdotp \cot^{2} x \ dx$

Answer 1

Solution:

To integrate the given function we must use integration by parts

Formula:

$\int U.V dx=U\int V dx-\int (\frac{dU}{dx}\int V dx)dx$

We know that $cot^{2}x= 1-cosec^{2}x$

$\int \:x\:cot^{2}x dx=\int x(1-cosec^{2}x) dx\\\\= \int\:x dx-\int\:x\:cosec^{2}xdx\\\\=\frac{x^{2}}{2}-(x\int cosec^{2}x dx- \int(\frac{d\:x}{dx}\int\:cosec^{2}x dx)dx)\\\\=\frac{x^{2}}{2}+xcot x- \int(cot x)dx\\\\=\frac{x^{2}}{2}+xcot x- log(sin x)+C$

So,

$\int\:x\:cot^{2}x dx=\frac{x^{2}}{2}+xcot x- log(sin x)+C$