The first step is to convert the cosine function to the exponential form using Euler's formula.
Cos(2x) = 1/2 * [e^(i2x)-e^(-i2x)]
The next step is to distribute the outside exponential that is 1/2*[e^(3x+i2x)-e^(3x-i2x)].
Now the integral becomes a very simple u and v substitution integral. That is let u = (3+i2)x and v = (3-i2)x therefore du = (3+i2)dx and dv = (3-i2)dx.
Is this far enough help or do you need the next steps?