# Definite Integral to Infinity of Exponential of -(a x^2 plus b over x^2)

## Theorem

$\displaystyle \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }$

where $a$ and $b$ are strictly positive real numbers.

## Proof

 $\displaystyle \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x$ $=$ $\displaystyle \int_0^\infty \map \exp {-a \paren {x^2 + \frac b {a x^2} } } \rd x$ $\displaystyle$ $=$ $\displaystyle \int_0^\infty \map \exp {-a \paren {\paren {x - \frac 1 x \sqrt {\frac b a} }^2 + 2 \sqrt {\frac b a} } } \rd x$ Completing the Square $\displaystyle$ $=$ $\displaystyle \map \exp {-2 \sqrt {a b} } \int_0^\infty \map \exp {-a \paren {x - \frac 1 x \sqrt {\frac b a} }^2} \rd x$ Exponential of Sum $\displaystyle$ $=$ $\displaystyle \frac 1 2 \map \exp {-2 \sqrt {a b} } \int_{-\infty}^\infty \map \exp {-a \paren {x - \frac 1 x \sqrt {\frac b a} }^2} \rd x$ Definite Integral of Even Function $\displaystyle$ $=$ $\displaystyle \frac 1 2 \map \exp {-2 \sqrt {a b} } \int_{-\infty}^\infty \map \exp {-a u^2} \rd u$ Glasser's Master Theorem $\displaystyle$ $=$ $\displaystyle \map \exp {-2 \sqrt {a b} } \int_0^\infty \map \exp {-a u^2} \rd u$ Definite Integral of Even Function $\displaystyle$ $=$ $\displaystyle \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }$ Definite Integral to Infinity of $\map \exp {-a x^2}$

$\blacksquare$