How is the power calculated for Si(t) in the picture?
The first image you posted appears to use Parseval's Relation after expressing the \$S_i(t)\$ as an exponential Fourier series.

Your method (the second image, I think) will produce the same answer, as long as you're careful about your limits of integration.  Let's assume that you've defined \$T_0 = \frac{2\pi}{\omega_c+\omega_m}\$.  Then your underlined terms can be recombined (using Euler "in reverse") to produce a cosine with frequency \$2(\omega_c+\omega_m)\$.  This cosine will complete two full oscillations in \$T_0\$ seconds, so its integral is zero.