Question

resolva integral por partes ∫ x In(3x) dx

Answer 1

Resolver a integral

$\int{x\mathrm{\,\ell n}\left(3x \right )\,dx}$

por partes.


$\begin{array}{ll} u=\mathrm{\ell n}\left(3x \right )\;\;&\;\;dv=x\,dx\\ \\ du=\dfrac{1}{\diagup\!\!\!\! 3x}\cdot \diagup\!\!\!\! 3\,dx\;\;&\;\;v=\dfrac{x^{2}}{2} \end{array}$


$\int{u\,dv}=uv-\int{v\,du}\\ \\ \\ \int{x\mathrm{\,\ell n}\left(3x \right )\,dx}=\dfrac{x^{2}}{2}\cdot \mathrm{\ell n}\left(3x \right )-\int{\dfrac{x^{2}}{2}\cdot \dfrac{1}{x}\,dx}\\ \\ \\ \int{x\mathrm{\,\ell n}\left(3x \right )\,dx}=\dfrac{x^{2}}{2}\cdot \mathrm{\ell n}\left(3x \right )-\dfrac{1}{2}\int{x\,dx}\\ \\ \\ \int{x\mathrm{\,\ell n}\left(3x \right )\,dx}=\dfrac{x^{2}}{2}\cdot \mathrm{\ell n}\left(3x \right )-\dfrac{1}{2}\cdot\dfrac{x^{2}}{2}+C\\ \\ \\ \boxed{\begin{array}{c} \int{x\mathrm{\,\ell n}\left(3x \right )\,dx}=\dfrac{x^{2}}{2}\cdot \mathrm{\ell n}\left(3x \right )-\dfrac{x^{2}}{4}+C \end{array}}$