The next three problems ask you to show that the definite integral on [0, 2] of sin(mx) multiplied by almost any other combination of sin(nx) or cos(nx) is 0. The only nonzero value comes when sin(mx) is multiplied by itself.

19. Show that if m and n are integers with m # n, then sin(mx).sin(nx) dx = 0. 2JT 20. Show that if m and n are integers, then sin(mx).cos(nx) dx =0. (Consider m = n and m # n.) 0 2JT 21. Show that if m + 0 is an integer, then sin(mx).sin(mx) dx = J. 0