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