Question Number 172010 by Mikenice last updated on 23/Jun/22
$${find}\:{integrate}: \\ $$$$\int{xe}^{{x}} {dx} \\ $$
Answered by puissant last updated on 23/Jun/22
![K=∫xe^x dx ; { ((u′=e^x )),((v=x)) :}⇒ { ((u=e^x )),((v′=1)) :} ⇒ K=[xe^x ]−∫e^x dx ⇒ K=e^x (x−1)+C](https://www.tinkutara.com/question/Q172049.png)
$${K}=\int{xe}^{{x}} {dx}\:\:;\:\:\begin{cases}{{u}'={e}^{{x}} }\\{{v}={x}}\end{cases}\Rightarrow\:\begin{cases}{{u}={e}^{{x}} }\\{{v}'=\mathrm{1}}\end{cases} \\ $$$$\Rightarrow\:{K}=\left[{xe}^{{x}} \right]−\int{e}^{{x}} {dx} \\ $$$$\Rightarrow\:{K}={e}^{{x}} \left({x}−\mathrm{1}\right)+{C} \\ $$