7.1 Integration By Parts/53

$y=xe^{-0.4x}~~~x=0,x=1$

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

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u &= -0.4x \\[2ex] du &= dx \\[2ex] \end{align} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left (-\frac{25}{2}e^{-2}-0 \right )+\frac{5}{4}\int_{5}^{0}e^{u} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{25}{2}e^{-2}+\left [ \frac{5}{4}\left ( e^{-0.4x} \right ) \right ]~~~=~~~ -\frac{25}{2}e^{-2}-\frac{25}{4}e^{-2}+\frac{25}{4} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\frac{75}{4}e^{-2}+ \frac{25}{4} }