Weak statement for Galerkin Method
 

I have the equation -∇^2u+u-x^2=0 where u(0)=u(1)=0.
I need to solve it using Galerkin Method using the approximation of psi=sin(n*pi*x). I've already solved the exact solution but when I work out the Galerkin method I only get half a sine wave. I believe I'm messing up the weak statement. Could somebody walk me through the method to construct a weak statement?