Definite Integral over Reals of Exponential of -(a x^2 plus b x plus c)

Theorem

 * $\displaystyle \int_{-\infty}^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} }$

where $a$, $b$ and $c$ are real numbers with $a > 0$.