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valentinak56 [21]
3 years ago
11

Find the average value of the function f(t)=cos13(5t)sin(5t) f(t)=cos13⁡(5t)sin⁡(5t) on the interval [4,10].

Mathematics
1 answer:
Vilka [71]3 years ago
6 0
The average value is given by

\displaystyle\frac1{10-4}\int_4^{10}\cos^{13}5t\sin5t\,\mathrm dt

Setting y=\cos5t, you get \mathrm dy=-5\sin5t\,\mathrm dt, and the integral becomes

\displaystyle-\frac1{30}\int_{\cos20}^{\cos50}y^{13}\,\mathrm dy
=-\dfrac1{30}\dfrac{y^{14}}{14}\bigg|_{y=\cos20}^{y=\cos50}
=-\dfrac{\cos^{14}50-\cos^{14}20}{420}\approx-0.00145
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