6.5 Average Value of a Function/5

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 f(t) = te^{-t^2} \quad [0, 5] }

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} f_{avg} &= \frac{1}{5}\int_{0}^{5}te^{-t^2}\,dx \\[2ex] & = \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) \\[2ex] & = \frac{1}{10}\int_{-25}^{0}e^u\,du \\[2ex] & = \frac{1}{10}e^u \bigg|_{-25}^{0} \\[2ex] & = \frac{1}{10}-\frac{1}{10}e^{-25} \\[2ex] & = \frac{1}{10}(1-e^{-25}) \\[2ex] \end{align} }

${\begin{aligned}&u=t^{2}\\[2ex]&{\frac {du}{dt}}=2t\\[2ex]&du=-2t\cdot {dt}\\[2ex]&-{\frac {1}{2}}du=t\cdot {dt}\\[2ex]\end{aligned}}$