Question

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

Answer 1

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