Hello Cameron,
you Mathematica inputs contains several errors: sin(x) -> Sin[x], pi -> [Pi]. Then you have to tell Mathematica about n,m being positive Integers. Find below the corresponding input and results (Mathematica 10.1):
In[208]:= Simplify[Integrate[ Sin[n \[Pi] x] Sin[ m \[Pi] y/2] (x + y), {x, 0, 1}, {y, 0, 2}], Assumptions -> {m > 0, n > 0, {m, n} \[Element] Integers}]
Out[208]= (2 (-2 (-1)^m + (-1)^(1 + n) + 3 (-1)^(m + n)))/(m n \[Pi]^2 )
Best regards,
Michael