Fourier series calculation
Hint: this is a lot easier if you use the Euler formula https://en.wikipedia.org/wiki/Euler%27s_formula

Then the integral simply becomes
$$X(n)=\frac{1}{L}\int_{0}^{2L}e^{(-a+i\omega)\cdot t}, a_{n}=real[X(n)], b_{n} = imag[X(n)]$$

which is pretty straight forward to solve