Question

Integrate the function: $\frac{(x+1)(x+log x)^2}{x}$

Answer 1

$\frac{(x+1)(x+log x)^2}{x} = \frac{x+1}{x}(x+log x)^2 = (1+\frac{1}{x})(x+log x)^2.$
$Let (x+logx) = t$
$∴ (1+\frac{1}{x})dx = dt$
$⇒ ∫(1+\frac{1}{x})(x+log x)^2 dx = ∫t^2dt$
$=\frac{t^3}{3}+C$
$=\frac{1}{3}(x+log x)^3 +C$