Question

Integrate the function.
$\int xlogxdx.$

Answer 1

Let $I=\int$ x log x dx. On taking log x as first function and x as second function and integrating by parts, we get
$(\because$ log function comes before algebraic function in ILATE)
$I=logx\int xdx-\int \left[\frac{d}{dx}\right(logx)\int xdx]dx=\frac{x^{2}logx}{2}-\frac{1}{2}\int \frac{1}{x}{x}^{2}dx=\frac{x^{2}logx}{2}-\frac{1}{2}\int xdx\Rightarrow I=\frac{x^{2}logx}{2}-\frac{1}{4}{x}^2+C$