How would you go about integrating a function which has an exponential and a cos/sin term?

For this type of integral you would need to use integration by parts twice. Remember this is where the integral of u*(dv/dx)=uv-(integral with respect to x)(v*(du/dx)). For example let's say you need to integrate sin(x)e^x, it doesn't matter which you choose to be u and dv/dx as sin(x) is cyclic and e^x doesn;t change under differnetiation/integration. And so arbitarily say u=sin(x) and dv/dx=e^x therefore du/dx=cos(x) and v=e^x. We will call the original integral I and so now we have I=sin(x)e^x-(integral with respect to x)(cos(x)e^x). then perform it again but with different variables. Now we change u=cos(x) and dv/dx=e^x, it can be usefull to use subscripts to not confuse u and v from what they previously were, therefore du/dx=-sin(x), v=e^x. From this we have I=sin(x)e^x-cos(x)e^x-(integral with respect to x)(sin(x)e^x)+c, and the integral left now is the same as the original and so equals I, hence this can be reaaranged to 2I=sin(x)e^x-cos(x)e^x+c. This can be cleaned up to get I=((sin(x)-cos(x))e^x)/2+c, is this all clear?