Question

Calcule as seguintes integrais indefinidas pelo método
de integração por partes:

∫ x ln x dx

Answer 1

$\int{x\,\mathrm{\ell n\,}x\,dx}$


Integrando por partes:

$\begin{array}{cc} u=\mathrm{\ell n\,}x\;&\;du=\dfrac{dx}{x}\\ \\ dv=x\,dx\;&\;v=\dfrac{x^{2}}{2} \end{array}\\ \\ \\ \\ \int{u\,dv}=uv-\int{v\,du}\\ \\ \\ \int{x\,\mathrm{\ell n\,}x\,dx}=\dfrac{x^{2}}{2}\,\mathrm{\ell n\,}x-\int{\dfrac{x^{2}}2\cdot \dfrac{dx}{x}}\\ \\ \\ \int{x\,\mathrm{\ell n\,}x\,dx}=\dfrac{x^{2}}{2}\,\mathrm{\ell n\,}x-\dfrac{1}{2}\int{x\,dx}\\ \\ \\ \int{x\,\mathrm{\ell n\,}x\,dx}=\dfrac{x^{2}}{2}\,\mathrm{\ell n\,}x-\dfrac{1}{2}\cdot \dfrac{x^{2}}{2}+C\\ \\ \\ \boxed{\begin{array}{c}\int{x\,\mathrm{\ell n\,}x\,dx}=\dfrac{x^{2}}{2}\,\mathrm{\ell n\,}x-\dfrac{x^{2}}{4}+C \end{array}}$