$ \Large I = \int_{0}^{\infty} e^{ax} cos(x) dx $

I calculated the indefinite integral and founded it equals $ \large \dfrac{e^{ax}}{a^{2}+x} (sin(x)+acos(x) ) + C $

Now, replacing $ \infty $ by t and evaluate the limit I faced this limit :

$ \large \dfrac{1}{a^{2}+1} \lim_{t \to \infty} e^{at} \cdot (sin(t)+a cos(t)) - a ) $

I stopped at this limit.