f ( t ) = t e − t 2 [ 0 , 5 ] {\displaystyle f(t)=te^{-t^{2}}\quad [0,5]}
f a v g = 1 5 ∫ 0 5 t e − t 2 d x = 1 5 ∫ 0 5 − 1 2 ( e u ) d u = 1 5 ∫ 0 25 e u ( − 1 2 d u ) = 1 10 ∫ − 25 0 e u d u = 1 10 e u | − 25 0 = 1 10 − 1 10 e − 25 = 1 10 ( 1 − e − 25 ) {\displaystyle {\begin{aligned}f_{avg}&={\frac {1}{5}}\int _{0}^{5}te^{-t^{2}}\,dx={\frac {1}{5}}\int _{0}^{5}-{\frac {1}{2}}(e^{u})\,du\\[2ex]={\frac {1}{5}}\int _{0}^{25}e^{u}(-{\frac {1}{2}}du)={\frac {1}{10}}\int _{-25}^{0}e^{u}\,du\\[2ex]={\frac {1}{10}}e^{u}{\bigg |}_{-25}^{0}={\frac {1}{10}}-{\frac {1}{10}}e^{-25}={\frac {1}{10}}(1-e^{-25})\\[2ex]\end{aligned}}}
u = t 2 d u d t = 2 t d u = − 2 t ⋅ d t − 1 2 d u = t ⋅ d t {\displaystyle {\begin{aligned}&u=t^{2}\\[2ex]&{\frac {du}{dt}}=2t\\[2ex]&du=-2t\cdot {dt}\\[2ex]&-{\frac {1}{2}}du=t\cdot {dt}\\[2ex]\end{aligned}}}
![{\displaystyle f(t)=te^{-t^{2}}\quad [0,5]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/821f8b378982a8352dcc9c3e024a32563df501e0)
![{\displaystyle {\begin{aligned}f_{avg}&={\frac {1}{5}}\int _{0}^{5}te^{-t^{2}}\,dx={\frac {1}{5}}\int _{0}^{5}-{\frac {1}{2}}(e^{u})\,du\\[2ex]={\frac {1}{5}}\int _{0}^{25}e^{u}(-{\frac {1}{2}}du)={\frac {1}{10}}\int _{-25}^{0}e^{u}\,du\\[2ex]={\frac {1}{10}}e^{u}{\bigg |}_{-25}^{0}={\frac {1}{10}}-{\frac {1}{10}}e^{-25}={\frac {1}{10}}(1-e^{-25})\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/98019fcf2fc0972ef92efd99edf25fcaee5bc014)
![{\displaystyle {\begin{aligned}&u=t^{2}\\[2ex]&{\frac {du}{dt}}=2t\\[2ex]&du=-2t\cdot {dt}\\[2ex]&-{\frac {1}{2}}du=t\cdot {dt}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5482a82b0829dc44071e1ab066305da32657604c)