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## Homework Statement

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

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

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 [itex]\frac{α}{ω}[/itex] << 1, and neglect quantities proportional to α[itex]^{2}[/itex].

## The Attempt at a Solution

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

But for the next part, I think they're asking me to integrate I[itex]^{2}[/itex]Rdt over the interval, and I am not seeing how to approach that integral. Expanding the equation for I to get I[itex]^{2}[/itex] 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?