Question

$\int \sqrt{x}{\left(\log x\right)}^{2}dx=\frac{2}{3}{x}^{3/2}[{\left(\log x\right)}^2-\frac{4}{3}\log x+\frac{k}{9}].$ Find $k$.

Answer 1

$I=\int \sqrt{x}{\left(\log x\right)}^{2}dx$
$=\frac{2}{3}{x}^{3/2}{\left(\log x\right)}^2-\frac{2}{3}\int 2\sqrt{x}\log xdx$
$=\frac{2}{3}{x}^{3/2}{\left(\log x\right)}^2-\frac{4}{3}\int \sqrt{x}\log xdx$
$=\frac{2}{3}{x}^{3/2}{\left(\log x\right)}^2-\frac{8}{9}{x}^{3/2}\log x+\frac{8}{9}\int \sqrt{x}dx$
$=\frac{2}{3}{x}^{3/2}{\left(\log x\right)}^2-\frac{8}{9}{x}^{3/2}\log x+\frac{16x^{3/2}}{27}$
$\Rightarrow k=8$