Question

Qual a resolução da integral
x³ex²dx

Answer 1

$I=\displaystyle\int{x^{3}\,e^{x^{2}}\,dx}\\ \\ \\ =\int{x^{2}\cdot x\,e^{x^{2}}\,dx}\\ \\ \\ =\int{x^{2}\cdot e^{x^{2}}\,x\,dx}$


Método de integração por partes:

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