Question

Integrate the following functions.
$\int \frac{(x+1)(x+logx{)}^2}{x}dx.$

Answer 1

$\int \frac{(x+1)(x+logx{)}^2}{x}dx=\int (x+logx{)}^2\left(\frac{x+1}{x}\right)dx=\int (x+logx{)}^2(1+\frac{1}{x})dx$
Let $x+logx=t\Rightarrow (1+\frac{1}{x})dx=dt$
$\therefore \int (1+\frac{1}{x})(x+logx{)}^{2}dx=\int t^{2}dt=\frac{t^3}{3}+C=\frac{(x+logx{)}^3}{3}+C$