Hello Dear Wolfram Community,
I am trying to compute the following definite integral in Mathematica:
Sqrt[u]/(e + u)
over du from 0 to infinity
where e>0
The expected result is
\Pi / Sqrt[e]
I have tried the following Mathematica inputs:
1.
Assuming[e > 0, Integrate[Sqrt[u] / (e + u), {u, 0, Infinity}, GenerateConditions -> False]]
This input does not produce a convergence error but returns the incorrect result:
-Sqrt[e] \[Pi]
where the incorrect scaling with Sqrt[e] is the main issue. I’m not worried about the sign.
Other approaches like:
Integrate[Sqrt[u] / (e + u), {u, 0, Infinity}, Assumptions -> {e > 0}, PrincipalValue -> True]
or
Integrate[Sqrt[u] / (e + u), {u, 0, Infinity}, Assumptions -> {e > 0}]
also fails to give the expected result, suggesting that Mathematica struggles with interpreting the integral’s convergence or symmetry. (Both give as Output, that the Integral doesn’t converge)
How can I adjust my approach to compute this integral correctly in Mathematica?
Any advice would be greatly appreciated!