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

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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


Proof

\(\ds \int_{-\infty}^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x\) \(=\) \(\ds \int_{-\infty}^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2 + \frac {b^2} {4 a} - c} \rd x\) Completing the Square
\(\ds \) \(=\) \(\ds \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_{-\infty}^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2} \rd x\) Exponential of Sum
\(\ds \) \(=\) \(\ds \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_{-\infty}^\infty \map \exp {-\paren {\sqrt a x + \frac b {2 \sqrt a} }^2} \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 {\sqrt a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_{-\infty}^\infty \map \exp {-u^2} \rd u\) substituting $u = \sqrt a + \dfrac b {2 \sqrt a}$
\(\ds \) \(=\) \(\ds \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} }\) Gaussian Integral

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definite_Integral_over_Reals_of_Exponential_of_-(a_x%5E2_plus_b_x_plus_c)&oldid=541935"