I am looking for the value of the following integral:

$$\int_u^\infty \sqrt{(x^2-a)} \exp\left({-\left(bx^2+\frac{c}{x^2}\right)}\right)\text dx$$

I encountered this problem when trying to find the expected value of a normal PDF where the standard deviation is a function of a Rayleigh random variable. I have seen integrals like $\int e^{-\left(bx^2+\frac{c}{x^2}\right)}\text dx$, or $\int e^{-\left(bx+\frac{c}{x}\right)}\text dx$ where the solutions involve closed form exponentials or modified Bessel functions, but I can't figure it out for the integral explained above.

  • $\begingroup$ what do we know about the variables? $\endgroup$ – Dr. Sonnhard Graubner May 8 '15 at 14:35
  • $\begingroup$ @Dr.SonnhardGraubner suppose we want to find $\int_0^\infty x e^{-x^2} (x \phi(d/\sqrt{e^2x^2+f^2}) ) dx$, where $\phi(.)$ is the standard normal PDF. I tried to reformulate it above format. $\endgroup$ – Alireza May 8 '15 at 14:56
  • $\begingroup$ @Dr.SonnhardGraubner assume that a,b,c>0 $\endgroup$ – Alireza May 8 '15 at 15:08
  • $\begingroup$ i think with $a=0$ it s possible $\endgroup$ – tired May 8 '15 at 15:17
  • $\begingroup$ @tired actually I need it for positive a, but a=0 solution may help :) $\endgroup$ – Alireza May 8 '15 at 15:19

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