Question

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

Answer 1

$I=\int {1\times \left(\log x\right)}^{2}dx$

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

Let $v=1$, now applying integration by-parts, we get

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

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

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

$dv=dx\Rightarrow v=x$

$I=x{\left(\log x\right)}^2-2[x\log x-\int x\times \frac{1}{x}dx]$

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

$I=x{\left(\log x\right)}^2-2[x\log x-x]+c$

$I=x{\left(\log x\right)}^2-2x\log x+2x+c$