An integral with cosine and exponential

Note that $\mathcal{L}\{\cos(6t)\}=\dfrac{s}{s^2+36}$ . Therefore, $\mathcal{L}\{t\cos(6t)\}=-\dfrac{\mathrm d}{\mathrm ds}\left(\dfrac{s}{s^2+36}\right)=\dfrac{s^2-36}{(s^2+36)^2}$ . Here, $s=1$ . So, $\displaystyle\int_0^\infty t\cos(6t)e^{-t}\mathrm dt=\dfrac{1-36}{37^2}=\dfrac{-35}{1369}$ .