Question

Integrate the function.
$\int x^{2}logxdx.$

Answer 1

On taking log x as first function and $x^2$ as second function and integrating by parts, we get
$(\because$ log function comes before algebraic function in ILATE)
$I=\int x^{2}logxdx=logx\int x^{2}dx-\int \left[\frac{d}{dx}\right(logx)\int x^{2}dx]dx=\frac{x^3}{3}logx-\int [\frac{1}{x}.\frac{x^3}{3}]dx=\frac{x^3}{3}logx-\frac{1}{3}\int x^{2}dx=\frac{x^3}{3}logx-\frac{x^3}{9}+C$