Convolution Integral of Harmonic Signal (Cosine) with the Sinc Function
As $T\rightarrow \infty$, the integral becomes the convolution integral. You can use the fact that convolution in the time domain is multiplication in the frequency domain and then you have a single carrier being passed through a low pass filter. The term where $\omega=\omega_c$ just has to do with how you define the ideal low pass filter and in this case you'd say that the value at exactly $\omega=\omega_c$ is equal to $\frac{1}{2}$.