7.1 Integration By Parts/53

$y=xe^{-0.4x}~~~y=0,x=5$

${\begin{aligned}\underbrace {\int _{0}^{5}xe^{-0.4x}dx} _{\begin{aligned}u&=x&dv&=e^{-0.4x}dx\\[0.6ex]du&=dx&v&={\frac {5}{2}}e^{-0.4x}\end{aligned}}\\[2ex]\end{aligned}}$

$\left[{\frac {5}{2}}e^{-0.4x}\right]{\bigg |}_{0}^{5}+\int _{0}^{5}{\frac {5}{2}}e^{-0.4x}dx$

${\begin{aligned}u&=-0.4x\\[2ex]du&=-0.4dx\\[2ex]\end{aligned}}$

$\left(-{\frac {25}{2}}e^{-2}-0\right)+{\frac {25}{4}}\int _{0}^{5}e^{u}$

$-{\frac {25}{2}}e^{-2}+\left[{\frac {25}{4}}\left(e^{-0.4x}\right)\right]~~~=~~~-{\frac {25}{2}}e^{-2}-{\frac {25}{4}}e^{-2}+{\frac {25}{4}}~~~=~~~-{\frac {75}{4}}e^{-2}+{\frac {25}{4}}$