The average value of a function over an interval [tex][a,b][/tex] is given by
[tex]\displaystyle\frac1{b-a}\int_a^bf(x)\,\mathrm dx[/tex]
We have
[tex]\displaystyle\frac1{\frac\pi2-0}\int_0^{\pi/2}e^{\sin t}\cos t\,\mathrm dt=\frac2\pi\int_0^{\pi/2}e^{\sin t}\,\mathrm d(\sin t)[/tex]
[tex]=\displaystyle\frac2\pi\int_{\sin0}^{\sin(\pi/2)}e^u\,\mathrm du=\frac2\pi e^u\bigg|_{\sin0}^{\sin(\pi/2)}[/tex]
[tex]=\dfrac{2(e^1-e^0)}\pi=\dfrac{2(e-1)}\pi[/tex]
