Your setup is correct.
You need to use integration by parts.
Now do the same thing to the other integral and simplify.
First, please forgive the poor notation used here as LaTeX does not seem to be working properly for the time being (http://www.mathhelpforum.com/math-help/f47/working-latex-issues-178319-2.html)?
This is a problem from "Elementary Differential Equations and Boundary Value Problems" by Boyce and DiPrima. Problem 10.7.1, one that should be trivial, I assume.
"Consider an elastic string of length L whose ends are held fixed. The string is set in motion with no initial velocity from an initial position u(x, 0) = f(x). Find the displacement u(x, t) for the given initial position f(x)."
f(x) = (2x)/L when 0 <= 0 <= L/2
f(x) = (2(L-x))/L when L/2 < x <= L
I eventually arrive at c_n = (2/L) * definite integral from 0 to L of (f(x) * sin((n*pi*x)/L)) dx
c_n = (2/L) * definite integral from 0 to L/2 of (((2x)/L * sin((n*pi*x)/L)) dx + (2/L) * definite integral from L/2 to L of ((2(L-x)/L) * sin((n*pi*x)/L)) dx)
How should I go about evaluating these integrals? I believe it has something to do with odd and even functions?
From the Student Solutions Manual I see they arrive at the same c_n prior to evaluating the integrals, and after they have evaluated them (this step is not elaborated at all) they arrive at:
c_n = (8/(n^2 * pi^2)*sin((n*pi)/2)