Bizarre Integral

Calculus Level 4

Given that \[ \int_0^\infty \sqrt{\frac{\theta}{2\pi x^3}}\exp\left(-\frac{\theta (x-\mu)^2}{2\mu^2 x}\right)\,dx=1, \] for \(\,\mu>0\,\) and \(\,\theta>0\,\). Then determine \[ \int_0^\infty \sqrt{\frac{x}{\pi}}\exp\left(1-\frac{x}{4}-\frac{1}{x}\right)\,dx. \]

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