Energy and power of the product of an exponential and sinusoidal signal
Let's write the signal $x(t)$ in the general form

$$x(t)=e^{-\alpha t}\cos(\omega_0t)u(t)\tag{1}$$

Note that its energy is given by

$$E_x=\int_{-\infty}^{\infty}|x(t)|^2dt=\int_{0}^{\infty}e^{-2\alpha t}\cos^2(\omega_0)tdt\tag{2}$$

Since this is a homework type problem I won't solve it for you but I'll give you some hints:


Expand the integrand of $(2)$ as
$$e^{-2\alpha t}\frac12(1+\cos(2\omega_0t))=\frac12e^{-2\alpha t}+\frac12e^{-2\alpha t}\frac12(e^{j2\omega_0t}+e^{-j2\omega_0t})$$
Now you have two integrals over basic exponential functions, which should be very straightforward to solve.
The final result should be
$$E_x=\frac{1}{4\alpha}+\frac{\alpha}{4(\alpha^2+\omega_0^2)}$$
If the energy $E_x$ is finite, what can you conclude about the power of $x(t)$?