Question

$\int \frac{{\left(\log x\right)}^2}{x}dx,x>0$

Answer 1

Consider the given integral,

Let, $y=\int \frac{{\left(\log x\right)}^2}{x}dx$

Put,

$dt=\frac{1}{x}dx$

$dx=xdt$

$y=\int \frac{t^2}{x}xdt$

$y=\int t^{2}dt$

$y=\frac{t^3}{3}+C$

where t=logx

$y=\frac{{\log x}^3}{3}+C$

Hence, this the answer.