Question

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

Answer 1

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`

`=-\frac1{30}(y^(14))/(14)\bigg|_(y=\cos20)^(y=\cos50)`

`=-(\cos^(14)50-\cos^(14)20)/(420)\approx-0.00145`