Question

Evaluate $\int \sqrt{x}\log xdx$ on $(0,\infty )$.

Answer 1

$\int \sqrt{x}\log xdx$

$=\log x\int \sqrt{x}dx-\int \frac{1}{x}\cdot \frac{2}{3}{x}^{3/2}dx$, Use by parts

$=\frac{2}{3}x\sqrt{x}\log x-\frac{2}{3}\int \sqrt{x}dx$

$=\frac{2}{3}x\sqrt{x}\log x-\frac{2}{3}\cdot \frac{2}{3}x\sqrt{x}+C$

$=\frac{2}{3}x\sqrt{x}\log x-\frac{4}{9}x\sqrt{x}+C$