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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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    This was part of a larger homework problem i$n$ 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.2011-11-29

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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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    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, $\int u^u \mathrm du$.2011-11-28
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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.