Integrating factor dy/dx+ycotx=2x+x^2cot x (x≠0) is :

Given differential equation:

$\frac{dy}{dx}+y\cot x=2x+x^{2}\cot x,\ (x\neq0)$

This is linear form: $\frac{dy}{dx}+P(x)y=Q(x)$ with $P(x)=\cot x$.

Integrating factor $=\exp\!\Big(\int P(x)\,dx\Big)=\exp\!\Big(\int \cot x\,dx\Big)$

$=\exp(\ln|\sin x|)=\sin x$

Integrating factor = $\sin x$