Question

$\int (\log x{)}^{2}dx$

Answer 1

$\int {\left(\log x\right)}^{2}dx$

Let $dv=dx\Rightarrow v=x$

$u={\left(\log x\right)}^2\Rightarrow du=2\log x\times \frac{dx}{x}$

$=x{\left(\log x\right)}^2-\int x\times 2\log x\times \frac{dx}{x}$

$=x{\left(\log x\right)}^2-2\int \log xdx$

Let $u=\log x\Rightarrow du=\frac{1}{x}dx$ and $dv=dx\Rightarrow v=x$

$=x{\left(\log x\right)}^2-2[x\log x-\int x\times \frac{dx}{x}]$

$=x{\left(\log x\right)}^2-2x\log x+2\int dx$

$=x{\left(\log x\right)}^2-2x\log x+2x+c$ where $c$ is the constant of integration.