Question

Integrate the function.
$\int x$ log 2x dx.

Answer 1

On taking log 2x as first function and x as second function and integrating by parts, we get
$(\because$ log function comes before algebraic function in ILATE)
$I=\int xlog2xdx=log2x\int xdx-\int \left[\frac{d}{dx}\right(log2x)\int xdx]dx=\frac{x^2}{2}log2x-\int [\frac{1}{2x}\times 2\times \frac{x^2}{2}dx]=\frac{x^{2}log2x}{2}-\frac{1}{2}\int xdx\Rightarrow I=\frac{x^2}{2}log2x-\frac{1}{4}{x}^2+C$