Uni-variate Gaussian Integration:

$$\int_{-\infty}^\infty \exp\left( - ( a x^2 + b x + c ) \right) \; dx = \sqrt{\frac{\pi}{a}} \exp\left( \frac{b^2 - 4ac}{4a} \right)$$ (13)

or in the finite case

$$\int_{0}^L \exp\left( - ( a x^2 + b x + c ) \right) \; dx = \sqrt... ...}{2} \ensuremath{\mathrm{erfc}}\left( \frac{b + 2La}{2\sqrt{a}} \right) \right)$$ (14)

where $\ensuremath{\mathrm{erfc}}$ is the complimentary error function, defined as

$$\ensuremath{\mathrm{erfc}}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} \exp(-t^2) \, dt$$