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))
