Question

Use integration by parts to verify

|(from -pi to +pi)cosmxcosnxdx = pi when m=n

and

|(from -pi to +pi)cosmxcosnxdx = 0 when m and n are not equal.

Answer 1

Assume that for all cases that both m and n are integers.

For m = n:
|(from -pi to +pi) cos^2(mx) dx =
= |(from -pi to +pi) (1/2)(1 + cos(2mx)) dx
= (1/2)(pi + pi) + (1/4){sin(2mpi) - sin(-2mpi)}
= pi

For m not equal n:
|(from -pi to +pi) cos(mx) cos(nx) dx =
= |(from -pi to +pi) (1/2)*{cos(m+n)x + cos(m-n)x} dx
= (1/2(m+n))*(sin(m+n)pi - sin(-(m+n)pi)) + (1/2(m-n))*(sin(m-n)pi - sin(-(m-n)pi))
= 0