Question

Usando o método de integração por partes: ∫ u.dv = u.v - ∫ v.du determine a integral:
∫ ln x . 3x² dx

Answer 1

$\int{\mathrm{\ell n}(x)\cdot 3x^{2}\,dx}\\ \\ \\ \begin{array}{ll} u=\mathrm{\ell n}(x)\;\;&\;\;dv=3x^{2}\,dx\\ \\ du=\dfrac{dx}{x}\;\;&\;\;v=x^{3} \end{array}\\ \\ \\ \int{u\,dv}=uv-\int{v\,du}\\ \\ \\ \int{\mathrm{\ell n}(x)\cdot 3x^{2}\,dx}=\mathrm{\ell n}(x)\cdot x^{3}-\int{x^{3}\cdot \dfrac{dx}{x}}$

$\int{\mathrm{\ell n}(x)\cdot 3x^{2}\,dx}=\mathrm{\ell n}(x)\cdot x^{3}-\int{x^{3}\cdot \dfrac{dx}{x}}\\ \\ \\ \int{\mathrm{\ell n}(x)\cdot 3x^{2}\,dx}=\mathrm{\ell n}(x)\cdot x^{3}-\int{x^{2}\,dx}\\ \\ \\ \boxed{ \begin{array}{c} \int{\mathrm{\ell n}(x)\cdot 3x^{2}\,dx}=\mathrm{\ell n}(x)\cdot x^{3}-\dfrac{x^{3}}{3}+C \end{array} }$