2nalfq8

2022-07-04

What is the average value of the function $f\left(t\right)=t{e}^{-{t}^{2}}$ on the interval [0,5]?

soosenhc

Expert

$\frac{1}{5-0}{\int }_{0}^{5}t{e}^{-{t}^{2}}dt=-\frac{1}{10}{\int }_{0}^{5}{e}^{-{t}^{2}}\left(-2t\right)dt$
$=-\frac{1}{10}\left[{e}^{-{t}^{2}}{\right]}_{0}^{5}$
$=-\frac{1}{10}\left({e}^{-25}-{e}^{0}\right)$
$=\frac{1}{10}\left(1-{e}^{-25}\right)$

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