Fourier Transform Problem - absolute value, time-saving tricks, etc
As you suggested, you could simply solve the integral using $\cos(\omega_0t)=(e^{j\omega_0t}+e^{-j\omega_0t})/2$. But as you've also noted, with this type of problems there is usually some smart way using known transform pairs. And what you suggested appears to me a very sane approach: convolve the known transform of $\cos(\omega_0t)$ with the known transform of $e^{-at}u(t)$ and you're done. I would recommend to you to cross-check your result by solving the integral, which in this case is also quite straight-forward. But I can assure you that your result looks good.