y = x e − 0.4 x y = 0 , x = 5 {\displaystyle y=xe^{-0.4x}~~~y=0,x=5}
∫ 0 5 x e − 0.4 x d x ⏟ u = x d v = e − 0.4 x d x d u = d x v = 5 2 e − 0.4 x {\displaystyle {\begin{aligned}\underbrace {\int _{0}^{5}xe^{-0.4x}dx} _{\begin{aligned}u&=x&dv&=e^{-0.4x}dx\\[0.6ex]du&=dx&v&={\frac {5}{2}}e^{-0.4x}\end{aligned}}\\[2ex]\end{aligned}}}
[ 5 2 e − 0.4 x ] 0 5 + ∫ 0 5 5 2 e − 0.4 x d x {\displaystyle \left[{\frac {5}{2}}e^{-0.4x}\right]_{0}^{5}+\int _{0}^{5}{\frac {5}{2}}e^{-0.4x}dx}
u = − 0.4 x d u = − 0.4 d x {\displaystyle {\begin{aligned}u&=-0.4x\\[2ex]du&=-0.4dx\\[2ex]\end{aligned}}}
( − 25 2 e − 2 − 0 ) + 25 4 ∫ 0 5 e u {\displaystyle \left(-{\frac {25}{2}}e^{-2}-0\right)+{\frac {25}{4}}\int _{0}^{5}e^{u}}
− 25 2 e − 2 + [ 25 4 ( e − 0.4 x ) ] = − 25 2 e − 2 − 25 4 e − 2 + 25 4 = − 75 4 e − 2 + 25 4 {\displaystyle -{\frac {25}{2}}e^{-2}+\left[{\frac {25}{4}}\left(e^{-0.4x}\right)\right]~~~=~~~-{\frac {25}{2}}e^{-2}-{\frac {25}{4}}e^{-2}+{\frac {25}{4}}~~~=~~~-{\frac {75}{4}}e^{-2}+{\frac {25}{4}}}
![{\displaystyle {\begin{aligned}\underbrace {\int _{0}^{5}xe^{-0.4x}dx} _{\begin{aligned}u&=x&dv&=e^{-0.4x}dx\\[0.6ex]du&=dx&v&={\frac {5}{2}}e^{-0.4x}\end{aligned}}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/91d4a6c6285a2e1341ff3df89604c6777da55fd6)
![{\displaystyle \left[{\frac {5}{2}}e^{-0.4x}\right]_{0}^{5}+\int _{0}^{5}{\frac {5}{2}}e^{-0.4x}dx}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9c998b01560293e0940d71c525c1887428208028)
![{\displaystyle {\begin{aligned}u&=-0.4x\\[2ex]du&=-0.4dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5db33c5878b9e558dbbb4067f4b16ff433ee0744)
![{\displaystyle -{\frac {25}{2}}e^{-2}+\left[{\frac {25}{4}}\left(e^{-0.4x}\right)\right]~~~=~~~-{\frac {25}{2}}e^{-2}-{\frac {25}{4}}e^{-2}+{\frac {25}{4}}~~~=~~~-{\frac {75}{4}}e^{-2}+{\frac {25}{4}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/2ff5ba25b01b0795367d80ea196351e8e499216a)