Primitive function/x^2 over (x cos x -sin x)^2/Example

We want to find a primitive function for the function

${}f(x)={\frac {x^{2}}{(x\cos x-\sin x)^{2}}}\,.$

We first determine the derivative of

${\frac {1}{x\cos x-\sin x}}.$

This is

${}-{\frac {\cos x-x\sin x-\cos x}{(x\cos x-\sin x)^{2}}}={\frac {x\sin x}{(x\cos x-\sin x)^{2}}}\,.$

Therefore, we write ${}f$ as a product ${}f(x)={\frac {x\sin x}{(x\cos x-\sin x)^{2}}}\cdot {\frac {x}{\sin x}}$. We apply integration by parts, where we integrate the first factor and differentiate the second factor. The derivative of the second factor is

${}{\left({\frac {x}{\sin x}}\right)}'={\frac {\sin x-x\cos x}{\sin ^{2}x}}\,.$

Hence, we have

${}{\begin{aligned}\int _{}^{}f(x)\,dx&={\frac {1}{x\cos x-\sin x}}\cdot {\frac {x}{\sin x}}-\int _{}^{}{\frac {1}{x\cos x-\sin x}}\cdot {\frac {\sin x-x\cos x}{\sin ^{2}x}}\,dx\\&={\frac {1}{x\cos x-\sin x}}\cdot {\frac {x}{\sin x}}+\int _{}^{}{\frac {1}{\sin ^{2}x}}\,dx\\&={\frac {1}{x\cos x-\sin x}}\cdot {\frac {x}{\sin x}}-\cot x.\end{aligned}}$