$S=\frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} \zeta(s+2) \Gamma(s)\cos(\frac{\pi s}{2})x^{-s} ds$
I understand there are poles at $s=-1,0,-2$
My answer is $S = \frac{\pi^2}{6} - \frac{x\pi}{2} + \frac{x^2}{4}$
Evaluating the residue at $s=-2$ gives $\frac{x^2}{4}$, and at $s=0$ gives $\frac{\pi^2}{6}$ but I can't seem to evaluate it at $s=-1$ to get the $- \frac{x\pi}{2}$ term