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I am trying to do the integration in this way. This is just an example but I want to do something like this, the output of the first integration will be in the second integration. I hope it is clear from this. Is there any way to do it?

f[r_] := Sqrt[r^2 + r^3];
p71[r_]:=r^2
a1 = 5.93999103;
g91 = { };
While[a1 <= 20,
  p31 = NIntegrate[f[r], {r, 5.93999103, a1}];
  p41 = p71[a1]*p31;
  
  AppendTo[g91, p41];
  a1 += 0.1];

p318[nu_] := 
  NIntegrate[((4*nu)/((Exp[(nu)/(g91)^(1/4)]) - 1)), {r, 
    5.939991023, \[Infinity]}];
Plot[p318[nu], {nu, 0.001, 10}]
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  • $\begingroup$ p71 is undefined $\endgroup$
    – yarchik
    Aug 25, 2022 at 14:21
  • $\begingroup$ @yarchik sorry for that. I edited it. $\endgroup$
    – AAA
    Aug 25, 2022 at 14:24
  • $\begingroup$ When I paste your code into a fresh notebook and change only the last line to Table[p318[nu],{nu,1,10}] I get a flood of error messages, most of which look like they are from dividing by zero. Is that what you see? Can you track down why you are getting those? $\endgroup$
    – Bill
    Aug 25, 2022 at 15:06
  • $\begingroup$ @Bill that is only because the first element in g91 is zero. If we ignore that and do the integration then still I am not getting anything. so I do not know if this is the correct way to do it. Since g91 is an array which has elements at every r and I want to integrate new function. $\endgroup$
    – AAA
    Aug 25, 2022 at 15:09
  • $\begingroup$ g91is a numerical list of numbers and occurs in your last integral Exp[(nu)/(g91)^(1/4)] . What could be the purpose of such a construct? $\endgroup$ Aug 25, 2022 at 15:11

1 Answer 1

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If I correctly understand your calculations, you are looking for a function a0= 5.93999103; F[a1]=a1^2 Integrate[f[r],{r,a0,a1}]?

Instead of While try NDSolve:

a0= 5.93999103;
f[r_] := Sqrt[r^2 + r^3];
D[a1^2 Integrate[f[r],{r,a0,a1}],a1]; (* F'[a1]*)

F'[a1] depends on F[a1]

ode=Derivative[1][F][a1] == a1^2 f[a1] + (2 F[a1])/a1     
g91 = NDSolveValue[{ode, F[a0] == 0}, F, {a1, a0, 20}] (* function g91[a1]*)
Plot[g91[a1], {a1, a0, 20}, AxesLabel -> {a1, "g91[a1]"}]

enter image description here

Knowing g91 it's possible (hopefully) to evaluate the second integral.

But second integral doesn't converge!

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