Convolution with sinusoids using convolution theorem
Hint: You have to combine the resulting complex exponentials into sine and cosine terms:

$$\sin x=\frac{e^{jx}-e^{-jx}}{2j}\\
\cos x=\frac{e^{jx}+e^{-jx}}{2}$$

If you use \$h(t)=e^{-at}u(t)\$, you should end up with the expression

$$y(t)=\frac{1}{a^2+25}\left[a\sin(5t)-5\cos(5t)+5e^{-at}\right]u(t)$$