Find the average value have of the function h on the given i

Find the average value h_ave of the function h on the given interval.

h(x) = 2 cos⁴x sin x,    [0, ]

h_ave =  

Solution

Avg value=1/b-a*h(x) dx evaluated from a to b.

(1/) 2cosx sinx dx, from x = 0 to

= (2/) cosx sinx dx

Let u = cosx, then du = -sinx dx. The boundaries x = 0 to become u = 1 to -1. In terms of u, the integral becomes:
(-2/) u du, from u = 1 to -1

= -2/(5) (u), from u = 1 to -1
= -2/(5) [-1 - 1] = -2/(5) (-2) = 4/(5)

average value = 4/(5)