Question

Find the value of the following integrals for all possible values ​​of n and m.

a)

r cos 0

b)

c)

where P represents the fundamental period of the trigonometric functions.

Answer 1

(a) integrals^p_0 cos (2 pi n/p x) cos (2 pi m/p x)dx = 1/2 integrals^p_0 2 cos (2 pi n/p x) cos (2 pi n/p x) dx = 1/2 integrals^p_0 cos (2 pi/p (m + n) x) + cos (2 pi/p (n - m) x) dx [Because cos (A + B) + cos (A - B) = 2 cos A cos B] = 1/2 [(sin 2 pi/p (m + n) x)^p_0/2 pi (m + n)/p + (sin pi (n - m)/p x/2 pi (n - m)/p)^p_0] = 1/2 [p/2 pi (m + n) [0 - 0] + p/2 pi (m - n) [0 - 0] = 0 (b) integrals^p_0 sin (2 pi n/p x) - sin (2 pi m/p x) dx = 1/2 integrals^p_0 2 sin n/p x) sin (2 pi m/p) dx = 1/2 integrals^p_0 cos (2 pi n/p x - 2 pi m/p x) - cos (2 pi n x/p + 2 pi m/p x) dx [Because cos (A - B) - cos (A - B) = 2 sin A sin B] = 1/2 integrals^p_0 - [cos [2 pi/p (n - m) x] - cos (2 pi/p (n + m) x)] dx.