Adding the Integrals

$$\begin{array}{rl}& {\int }_{\mathrm{\infty }}^c{r}^{s-1}{e}^{-\pi is}g(-r)dr+{\int }_c^{\mathrm{\infty }}{r}^{s-1}{e}^{\pi is}g(-r)dr\\ & ={\int }_{\mathrm{\infty }}^c{r}^{s-1}{e}^{-\pi is}g(-r)dr+{\int }_c^{\mathrm{\infty }}{r}^{s-1}{e}^{\pi is}g(-r)dr\\ & =-{\int }_c^{\mathrm{\infty }}{r}^{s-1}{e}^{-\pi is}g(-r)dr+{\int }_c^{\mathrm{\infty }}{r}^{s-1}{e}^{\pi is}g(-r)dr\\ & =e^{\pi is}{\int }_c^{\mathrm{\infty }}{r}^{s-1}g(-r)dr-e^{-\pi is}{\int }_c^{\mathrm{\infty }}{r}^{s-1}g(-r)dr\\ & =(e^{\pi is}-e^{-\pi is}){\int }_c^{\mathrm{\infty }}{r}^{s-1}g(-r)dr\\ & =2i\sin (\pi s){\int }_c^{\mathrm{\infty }}{r}^{s-1}g(-r)dr.\end{array}$$