I'm having trouble finding out how to directly integrate the function $f(t)$ because of the $\cos^2(2t)$ term. I understand that $\cos^2(2t) = \frac{1}{2} + \frac{1}{2}\cos(4t)$ but I don't understand how this simplifies the problem so that
$\int_0^\infty {e^{-st}} + \int_0^\infty {e^{-st}(\frac{1}{2} + \frac{1}{2}\cos(4t))}$
Is an easier integral to solve.