Energy Conservation in RLC circuit

Homework Statement

Assume the current in a series RLC circuit is given by I = ACω(sin(ωt) + $\frac{α}{ω}$cos(ωt))e$^{-αt}$.

Calculate the energy stored in the circuit at t=0. Then calculate the energy stored in the circuit one-quarter cycle later, at t=$\frac{\pi}{2ω}$.

Verfiy that the difference is equal to the energy dissipated in the resistor R during this interval.

For this problem, assume the damping is slight, that is, that $\frac{α}{ω}$ << 1, and neglect quantities proportional to α$^{2}$.

The Attempt at a Solution

I'm pretty sure I have the first part right, since it seems like a straightforward use of $\frac{CV^{2}}{2}$ + $\frac{LI^{2}}{2}$. A couple of e$^{-αt}$'s simplify to 1 when you plug in 0 and $\frac{\pi}{2ω}$ for t since $\frac{α}{ω}$ << 1.

But for the next part, I think they're asking me to integrate I$^{2}$Rdt over the interval, and I am not seeing how to approach that integral. Expanding the equation for I to get I$^{2}$ just makes a mess no matter what small terms I ignore. I don't know if I'm just being clumsy with my math or if I'm approaching it wrong. I even tried using complex numbers to represent the power but I ended up with a nonsensical answer. Any advice on tackling this part of the problem?

Well, after expanding, I get I$^{2}$R = A$^{2}$C$^{2}$ω$^{2}$R(e$^{-2αt}$sin$^{2}$(ωt) + $\frac{2α}{ω}$e$^{-2αt}$sin(ωt)cos(ωt) + $\frac{α^{2}}{ω^{2}}$e$^{-2αt}$cos$^{2}$(ωt))