Integration

My understanding of " mean value of a function"
is [ ∫ from a to b f(x)dx]/(b - a).
Now f(x) = x - x cos6x. a = 0 and b = 7π.
So mean value = (1/7π) ∫ (x - x cos 6x) dx = (1/7π)[ ∫ x dx - ∫ x cos 6x dx].
From the hint given :
d/dx (1/6)x sin6x = (1/6)sin6x + x cos6x
d (1/6)x sin6x = (1/6)sin6x dx + x cos6x dx
∫ d[(1/6)x sin6x] = ∫(1/6)sin6x dx + ∫x cos6x dx
so I = ∫ xcos6x dx = ∫d[(1/6)xsin6x] - ∫(1/6)sin6x dx
= (1/6)xsin6x + (1/36)cos6x
For x = 0, I = 1/36.
For x = 7π, I = 1/36
So I = 0 from x = 0 to x = 7π.
∫ x dx from x = 0 to x = 7π = (7π)^2/2
So mean value of the density function = (1/7π) x (7π^2)/2 = 7π/2.