Question

Evaluate $\int x\ln \sqrt{xdx}$

Answer 1

$\int x\ln \sqrt{x}dx=\frac{1}{2}\int x\ln xdx$

$integratingbyparts$

$=\frac{1}{2}[\ln x\int xdx-\int \frac{d\left(\ln x\right)}{dx}(\int xdx)dx]=\frac{1}{2}[\frac{x^2\ln x}{2}-\int \frac{1}{x}\times \frac{x^2}{2}dx]$

$=\frac{x^2\ln x}{4}-\frac{x^2}{8}$