Question

qual a ∫ x e-²× dx $integral de x elevado a menos dois x dx$

Answer 1

$\displaystyle\int{x\,e^{-2x}\,dx}$


Método de integração por partes:

$\begin{array}{ll} u=x~~&~~du=dx\\ \\ dv=e^{-2x}\,dx~~&~~v=-\dfrac{1}{2}\,e^{-2x} \end{array}$


$\displaystyle\int{u\,dv}=uv-\int{v\,du}\\ \\ \\ \int{x\,e^{-2x}\,dx}=-\dfrac{1}{2}\,x\,e^{-2x}-\int{\left(-\dfrac{1}{2}\,e^{-2x} \right )\,dx}\\ \\ \\ \int{x\,e^{-2x}\,dx}=-\dfrac{1}{2}\,x\,e^{-2x}+\dfrac{1}{2}\cdot \int{e^{-2x}\,dx}\\ \\ \\ \int{x\,e^{-2x}\,dx}=-\dfrac{1}{2}\,x\,e^{-2x}+\dfrac{1}{2}\cdot \left(-\dfrac{1}{2}\,e^{-2x} \right )+C\\ \\ \\ \boxed{\begin{array}{c}\displaystyle\int{x\,e^{-2x}\,dx}=-\dfrac{1}{2}\,x\,e^{-2x}-\dfrac{1}{4}\,e^{-2x}+C \end{array}}$