This problem concerns application of a residue theorem. I am not sure if I can apply the residue theorem. Need some pointers.

Question

I am to evaluate the integral $$ \int\limits^{\infty}_{0}\frac{\sinh(y)}{y^{2}-c^{2}}dy. $$

What I am not sure is if I can apply the residue theorem. The function $\sinh(y)$ diverges at $y\!=\!\infty$. With a divergent integrand, how can I apply the residue theorem? The $\sinh(y)$ has an essential singularity at infinity.

Please let me know if I can apply the residue theorem even if the integrand diverges at infinity.