In a physics textbook I am working with, the following integral is caluclated:
Define the complex elastic modulus as $\overline{M} = M_1 + i M_2$ where $\overline{M}$ is a function of the angular frequency $\overline{M}(\omega)$. If $\epsilon = \epsilon_0 e^{i \omega t}$ and $\sigma = \overline{M} \epsilon$, the energy disspated in one cycle is
$$\Delta W = \int_{\omega t = 0}^{2 \pi} \sigma d \epsilon = \pi M_2 \epsilon_{0}^{2}$$
I don't quite understand how this integral is calculated. When I try to do this by myself I get:
$$\Delta W = \int_{\omega t = 0}^{2 \pi} \sigma d \epsilon = \int_{\omega t = 0}^{2 \pi} \overline{M} \epsilon d \epsilon = \left[\frac{\overline{M} \epsilon^2}{2} \right]_{\omega t = 0}^{2 \pi}$$
$$= \frac{\overline{M} (2 \pi)^2}{2} - \frac{\overline{M} \epsilon_0 ^{2}}{2} = 2 \pi^2 \overline{M} - \frac{\overline{M} \epsilon_0 ^{2}}{2}$$
This is as far as I get. I don't see how we can get the final result by plugging in $\overline{M} = M_1 + i M_2$, so obviously I'm doing something wrong. If anyone can show me how to actually compute this integral, I would be really grateful!