Question

Usando o método de integração por partes: $\int\limits u.dv = u.v - \int\limits v. du$
Determine a integral: $\int\limits ln           x. 3 x^{2} dx$

Answer 1

$I=\displaystyle\int{\mathrm{\ell n\,}x\cdot 3x^{2}\,dx}$

Método de integração por partes:

$\begin{array}{lcl} u=\mathrm{\ell n\,}x&\Rightarrow&du=\dfrac{1}{x}\,dx\\ \\ dv=3x^{2}\,dx&\Leftarrow&v=x^{3} \end{array}\\ \\ \\ \\ \displaystyle\int{u\,dv}=uv-\int{v\,du}\\ \\ \\ \int{\mathrm{\ell n\,}x\cdot 3x^{2}\,dx}=x^{3}\,\mathrm{\ell n\,}x-\int{x^{3}\cdot \dfrac{1}{x}\,dx}\\ \\ \\ \int{\mathrm{\ell n\,}x\cdot 3x^{2}\,dx}=x^{3}\,\mathrm{\ell n\,}x-\int{x^{2}\,dx}\\ \\ \\ \boxed{\begin{array}{c} \displaystyle\int{\mathrm{\ell n\,}x\cdot 3x^{2}\,dx}=x^{3}\,\mathrm{\ell n\,}x-\dfrac{x^{3}}{3}+C \end{array}}$