Question

Integrate the function w.r.t x : $x{\left(\log x\right)}^2$

Answer 1

Putting $logx=m$

$\frac{dx}{x}=dm$

$dx=e^{m}dm$

$\int m^2{e}^{2m}dm$

Integrating by parts we get,

$u=m^2,v=e^{2m}$

$u^{{}^{\prime }}=2m,\int vdm=\frac{1}{2}{e}^{2m}$

$⟹\frac{m^2{e}^{2m}}{2}-\int m{e}^{2m}dm$

Again integrating by parts

$u=m,v=e^{2m}$

$u^{{}^{\prime }}=1,\int vdm=\frac{e^{2m}}{2}$

$⟹\frac{m^2{e}^{2m}}{2}-\frac{m{e}^{2m}}{2}+\frac{e^{2m}}{4}$

$⟹\frac{(logx{)}^2{x}^2}{2}-\frac{x^{2}logx}{2}+\frac{x^2}{4}$