Make sure you using the right Mathematica syntax:Integrate[E^(7 I x). . . , Sin[2x], etc.
Here's the code that appears to work:
In[11]:= Integrate[E^(7 I x)*((E^(2 I x) + E^(-2 I x))/2), x] Out[11]= -(1/90) I E^(5 I x) (9 + 5 E^(4 I x))
Try replacing "i" with an undefined variable and then integrate. You can probably have Wolfram|Alpha do these integrals and give you the step by step instructions on how to solve them. After that, substitute the "i" back in the equation and do some algebraic simplification.
Doing integrals with complex numbers is not significantly more complex than doing them with real values.
