Question

Integrate with respect to $x$:
$xlnx$

Answer 1

Consider the given integral.

$I=\int x\ln xdx$

Since, $\int uvdx=u\int vdx-\int (\frac{d}{dx}\left(u\right)\int vdx)dx$

Therefore,

$I=\ln x\left(\frac{x^2}{2}\right)-\int \frac{1}{x}\left(\frac{x^2}{2}\right)dx$

$I=\frac{x^2}{2}\ln x-\frac{1}{2}\int xdx$

$I=\frac{x^2}{2}\ln x-\frac{1}{2}\left(\frac{x^2}{2}\right)+C$

$I=\frac{x^2}{2}\ln x-\frac{x^2}{4}+C$

Hence, this is the answer.