Question

Resolver pelo método de integração por partes

Answer 1

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


Método de integração por partes:

$\begin{array}{lcl} u=x&~\Rightarrow~&du=dx\\\\ dv=e^{3x}\,dx&~\Leftarrow~&v=\dfrac{1}{3}\,e^{3x} \end{array}\\\\\\\\ \displaystyle \int u\,dv=uv-\int v\,du\\\\\\ \int x\,e^{3x}\,dx=\dfrac{1}{3}\,x\,e^{3x}-\int \dfrac{1}{3}\,e^{3x}\,dx\\\\\\ \int x\,e^{3x}\,dx=\dfrac{1}{3}\,x\,e^{3x}-\dfrac{1}{3}\int e^{3x}\,dx\\\\\\ \int x\,e^{3x}\,dx=\dfrac{1}{3}\,x\,e^{3x}-\dfrac{1}{3}\cdot \left(\dfrac{1}{3}\,e^{3x} \right )+C\\\\\\ \therefore~~\boxed{\begin{array}{c}\displaystyle \int x\,e^{3x}\,dx=\dfrac{x\,e^{3x}}{3}-\dfrac{e^{3x}}{9}+C \end{array}}$