Replace
:
Split the integral at x = 1, and consider the latter one over [1, ∞) in which we replace
:
Then the original integral is equivalent to
Recall that for |x| < 1,
so that we can expand the integrand, then interchange the sum and integral to get
Integrate by parts, with
Recall the Fourier series we used in an earlier question [27217075]; if
where 0 ≤ x ≤ 1 is a periodic function, then
Evaluate f and its Fourier expansion at x = 1/2 :
So, we conclude that