Indefinite integral: chance of analytic solution? Is there a chance the following indefinte integral has an analytic solution? I(a, b) =int dx sqrt(a^x +b^x)

Wyatt Weeks

Wyatt Weeks

Answered question

2022-10-14

Indefinite integral: chance of analytic solution?
Is there a chance the following indefinte integral has an analytic solution?
I ( a , b ) = d x a x + b x
My attempts and those of Mathematica proved fruitless thus far...
Any suggestions for methods of attack are most welcome.

Answer & Explanation

Cagliusov8

Cagliusov8

Beginner2022-10-15Added 15 answers

Let α = 2 log ( b / a ) log a . Then Using the substitution t = a x / 2 we find that
I ( a , b ) = 2 log a 1 + t α d t .
I highly doubt that we have a general closed form for this. For example, this reduces to an elliptic integral when α = 3, which is obviously non-elementary.

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