Re: Solving special exponential integral

• To: mathgroup at smc.vnet.net
• Subject: [mg132308] Re: Solving special exponential integral
• From: "Alexander Elkins" <alexander_elkins at hotmail.com>
• Date: Fri, 7 Feb 2014 08:07:27 -0500 (EST)
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• References: <lckihv\$1hc\$1@smc.vnet.net>

```Using the variable substitution a -> Sqrt[x - b]
represented here by the function g, Mathematica gives
the following result:

In[1]:= With[{f =
Function[a, E^(-((a^2 * b * c^2)/(a^2 + b)))/(a^2 + b)^2],
g = Function[x, Sqrt[x - b]]},
Integrate[
f[g[x]] D[g[x], x], {x,
Sequence @@ InverseFunction[g] /@ {0, Infinity}}]]

Out[1]= ConditionalExpression[(
E^(-((b c^2)/
2)) \[Pi] (BesselI[0, (b c^2)/2] + BesselI[1, (b c^2)/2]))/(
4 b^(3/2)), Re[c^2] < 0 && b > 0]

Perhaps this helps...

"simone8888" <stefanvuckovic1 at gmail.com> wrote in message
news:lckihv\$1hc\$1 at smc.vnet.net...
> I have tried to solve this integral:
>
> Integrate[E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2,a]
> Mathemathica is not able to solve it, I have tried the integration by
parts and it did not work, as well as some substitutions. Any idea how to
tackle this problem?
>

```

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