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

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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 $\map \exp {-a x^2}$

$\blacksquare$


Sources