Bizarre Integral

Calculus Level 5

Given that 0θ2πx3exp(θ(xμ)22μ2x)dx=1, \int_0^\infty \sqrt{\frac{\theta}{2\pi x^3}}\exp\left(-\frac{\theta (x-\mu)^2}{2\mu^2 x}\right)\,dx=1, for μ>0\,\mu>0\, and θ>0\,\theta>0\,. Then determine 0xπexp(1x41x)dx. \int_0^\infty \sqrt{\frac{x}{\pi}}\exp\left(1-\frac{x}{4}-\frac{1}{x}\right)\,dx.

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