I was asked to show that this convolution integral
With $u(t)=\cos(\omega t)$, compute $y(t)=g(t)*\cos(\omega t)$ directly by evaluating
$$y(t) = \lim_{T\to\infty} \int_{-T}^T \frac{\sin(\omega_c \tau)}{\pi\tau}\cos(\omega(t-\tau))\ \mathsf d\tau $$
to show $$ y(t) = \begin{cases} \cos(\omega t),& \text{ for } |\omega|<\omega_c \\\ \frac12\cos(\omega t),& \text{ for } \omega=\omega_c \\\ 0,& \text{ for } |\omega|>\omega_c. \end{cases} $$
results in the answers also given in the image. Not quite sure how to approach this integral, everything seems to be coupled together.
Does anyone know how this kind of integrals is solved?