Question

Integrate the function.
$\int xse{c}^{2}xdx.$

Answer 1

Let $I=\int xse{c}^{2}xdx.$
On taking x as first function and $se{c}^2$ x as second function and integrating by parts, we get
$I=x\int se{c}^{2}xdx-\int \left[\frac{d}{dx}\right(x)\int se{c}^{2}xdx]dx$
$=xtanx-\int tanxdx=xtanx-log|secx|+C\Rightarrow I=xtanx+log|cosx|+C[log|secx|=log\left|\frac{1}{cosx}\right|=log1-log|cosx|=-log|cosx|(\because log=0\left)\right]$