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I want to calculate this integral but Mathematica does not solve it. Why? Can it be solved by Mathematica?

Integrate[Exp[-(Sqrt[1 + x^2] - 1)/t]*x^2, {x, 0, Infinity}]

I asked Mathematica to change variables to

 "u == Sqrt[1 + x^2] - 1"

and the result is Bessel function. but I'm not sure if

"IntegrateChangeVariables"

works properly.

This is the result:

ConditionalExpression[Exp^(1/t) t BesselK[2, 1/t], Re[t] > 0]

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  • $\begingroup$ Perhaps you have a typo... t-->x? $\endgroup$ Sep 3, 2023 at 14:46
  • $\begingroup$ @OkkesDulgerci no, it's t $\endgroup$
    – lia
    Sep 3, 2023 at 14:56
  • $\begingroup$ Haha so this is a little awkward, I did exactly what you did for the add-on edit to your post in your question, and posted it at the exact same time you made the edit. I did some numerical confirmation of the result for explicit values of $t$ at the bottom of my answer though to confirm ConditionalExpression[Exp^(1/t) t BesselK[2, 1/t], Re[t] > 0] is in fact equal to the integral $\endgroup$
    – ydd
    Sep 3, 2023 at 15:36
  • 1
    $\begingroup$ Thank you so much for your help $\endgroup$
    – lia
    Sep 3, 2023 at 15:43

1 Answer 1

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You just need to change variables. Using $u=\sqrt{x^2+1}$ appears to work:

int = Exp[-(Sqrt[1 + x^2] - 1)/t]*x^2;

new = IntegrateChangeVariables[
  Inactive[Integrate][int, {x, 0, Infinity}], u, u == Sqrt[1 + x^2]]

(*Inactive[Integrate][
 E^((1 - u)/t) u Sqrt[-1 + u^2], {u, 1, \[Infinity]}]*)

Yields $$\int _1^{\infty }u \sqrt{u^2-1} e^{\frac{1-u}{t}}du$$

Activating the output:

Activate[new]
(*ConditionalExpression[E^(1/t) t BesselK[2, 1/t], Re[t] > 0]*)

So $$\int_0^{\infty } x^2 \exp \left(-\frac{\sqrt{x^2+1}-1}{t}\right) \, dx = e^{1/t} ~t~ K_2\left(\frac{1}{t}\right)$$

for $\Re(t)>0$, where $K_n(t)$ is the modified Bessel function of the second kind.

Looking experimentally with values of $t$, we see this is correct

Table[NIntegrate[int, {x, 0, Infinity}], {t, 5}]
(*{4.41677, 24.8963, 73.3853, 161.878, 302.373}*)

Table[(E^(1/t) t BesselK[2, 1/t]), {t, 5}] // N

(*{4.41677, 24.8963, 73.3853, 161.878, 302.373}*)

```
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