# Help with finding integral

I've been trying for an embarrassingly long time to figure this one out. It looks like it should crack under integration by parts and integration by substitution, but I am having trouble with it. Any pointers?

$$\int {\exp(a \sqrt{x^2 + b} ) \over \sqrt{x^2+b} } dx$$

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You're sure there's no x in the numerator? – user7530 Nov 28 '11 at 11:21
Wolfram Alpha hasn't found result in the terms of standard mathematical functions.. – pedja Nov 28 '11 at 11:24
Wolfram Alpha says it has no expression in elementary terms. – lhf Nov 28 '11 at 11:26
This was part of a larger homework problem in a random signals class where I was trying to prove/disprove independence of two RVs with a joint pdf that looks similar to the integral above. Apparently solving it by directly using integration is not the right approach. – Mark Borgerding Nov 29 '11 at 16:20

The special case $a=1,b=0$ simplifies to

$$\int \frac{e^x}{x} \, dx$$

Which is known as the Exponential Integral where no closed form is known. Therefore your integral has no general closed form either.

Note: By closed form I mean that it is not expressible in terms of elementary functions.

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...I consider the exponential integral as a closed form expression in its own right. Better to say that the exponential integral is not expressible in terms of the elementary functions... – J. M. Nov 28 '11 at 13:05
I added a note to clarify what I mean, although if you consider $\text{Ei(x)}$ to be a closed form expression you could define every integral as a function and call it closed form :-) – Listing Nov 28 '11 at 13:09
Sure, but another part of my "closed-form" schtick is that there's a whole corpus of identities relating it to a lot of other functions. ;) You don't have that with, say, $\int u^u \mathrm du$. – J. M. Nov 28 '11 at 13:31

The natural substitution is to let $u=a\sqrt{x^2+b}$ so that $du=\frac{ax}{\sqrt{x^2+b}}$ and $x^2=\frac{u^2}{a^2}-b$. Our integral then is

$$\int \frac{e^u}{\sqrt{u^2-ba^2}} du.$$ However, there is no way to deal with something of this form. Are you sure that you are not missing a multiple of $x$? In other words, I think the question should be to integrate $$\int \frac{x\exp(a\sqrt{x^2+b})}{\sqrt{x^2+b}}dx$$ because then it gives $$\frac{1}{a}e^{a\sqrt{x^2+b}}$$ as the anti derivative.

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