\(\int_{t_{0}}^{t_{f}}{\text{erfc}\left(\frac{G}{\sqrt{t}}\right)\text{\ dt}}=\left(2\ G^{2}+t_{f}\right)\text{\ erfc}\left(\frac{G}{\sqrt{t_{f}}}\right)-\frac{2\ G\ \sqrt{t_{f}}\ e^{-\frac{G^{2}}{t_{f}}}}{\sqrt{\pi}}\ -2\ G^{2}-\left[\left(2\ G^{2}+t_{0}\right)\text{\ erfc}\left(\frac{G}{\sqrt{t_{0}}}\right)-\frac{2\ G\ \sqrt{t_{0}}\ e^{-\frac{G^{2}}{t_{0}}}}{\sqrt{\pi}}\ -2\ G^{2}\right]\)
\(=\left(2\ G^{2}+t_{f}\right)\text{\ erfc}\left(\frac{G}{\sqrt{t_{f}}}\right)-\frac{2\ G\ \sqrt{t_{f}}\ e^{-\frac{G^{2}}{t_{f}}}}{\sqrt{\pi}}-2\ G^{2}-\left(2\ G^{2}+t_{0}\right)\text{\ erfc}\left(\frac{G}{\sqrt{t_{0}}}\right)+\frac{2\ G\ \sqrt{t_{0}}\ e^{-\frac{G^{2}}{t_{0}}}}{\sqrt{\pi}}+2\ G^{2}\)
\(=\left(2\ G^{2}+t_{f}\right)\text{\ erfc}\left(\frac{G}{\sqrt{t_{f}}}\right)-\left(2\ G^{2}+t_{0}\right)\text{\ erfc}\left(\frac{G}{\sqrt{t_{0}}}\right)-\frac{2\ G\ \sqrt{t_{f}}\ e^{-\frac{G^{2}}{t_{f}}}}{\sqrt{\pi}}+\frac{2\ G\ \sqrt{t_{0}}\ e^{-\frac{G^{2}}{t_{0}}}}{\sqrt{\pi}}\ \) (14)