If you had done some theory of complex numbers this would be trivial, but

as we must assume otherwise you can do this using integration by parts.

I = Integral e^xcos(2x)dx

put dv=e^x and u=cos(2x), the the integration by parts rule gives:

I = e^x cos(2x) + 2 integral e^x sin(2x) dx

Now we do this again with the integral on the right, with dv=e^x and u=sin(2x),

to get:

I = e^x cos(2x) + 2 [e^x sin(2x) - 2 integral e^x cos(2x) dx]

or:

I = e^x cos(2x) + 2 e^x sin(2x) - 4 I,

so:

5I = e^x cos(2x) + 2 e^x sin(2x)

or:

I = [e^x cos(2x) + 2 e^x sin(2x)]/5

and finally add in a constant of integration:

I = [e^x cos(2x) + 2 e^x sin(2x)]/5 + C

RonL