Question

Evaluate: $\int \frac{\left(x+1\right)\left(x+\log x\right)}{x}dx$

Answer 1

$\Rightarrow \int \frac{(x+1)(\log x+x)}{x}dx$

Let $\log x+x=z$

$\Rightarrow (\frac{1}{x}+1)dx=dz$

$\Rightarrow \int zdz$

$\Rightarrow \frac{x+1}{x}dx=dz$

$\Rightarrow \frac{z^2}{2}+c$

Putting $z=x+\log x$

$I=\frac{(x+\log x{)}^2}{2}+c$