How to show that the inverse Gaussian density integrates to 1? How to prove
$\int_{0}^{\infty}\left[\frac{\lambda}{2\pi x^3}\right]^{1/2}\exp\left\{\frac{-\lambda(x-\mu)^2}{2\mu^2 x}\right\}dx=1$?
You can use the closed form formula for the more general integral


  $$\int_{ 0 }^{\infty} (ax^m)^{s} e^{\frac{-b(x-\mu)^2}{x}} dx = 2\,{a}^{s}{\mu}^{ms+1}{{\rm e}^{2b\mu}}
{{\rm K_{m s+1}}\left(2\,b\mu\right)},$$


where $\rm{K}_{\nu}(x)$ is the modified Bessel function  of the second kind.
Remember that google is your friend. Here are some sources where you can find the proof

Youtube:

http://www.youtube.com/watch?v=g_bCDcNWcgU

Math StackExchange:

How to show the normal density integrates to 1?

Wikipedia:

http://en.wikipedia.org/wiki/Gaussian_integral