Is this indefinite integral really unsolvable?

[Metaleer]

Hey, all.

Anyway, browsing the Internet a bit I found this integral:

\int \sqrt{1 + \frac{\ln x}{x}}dx​

as a proposed problem in a compilation of maths problems, as an integral from the MATYC journal. I gave it to Mathematica and WolframAlpha and they weren't able to solve it; WolframAlpha even claims there's no result found in terms of standard mathematical functions.

Any ideas? Perhaps Mathematica/WolframAlpha is on the fritz? :S

 

[micromass]

I highly suspect that wolfram alpha is correct. I think that there are indeed no elementary function that give that function as derivative. This isn't special, you know? It happens to a lot of integrals, for example

\int{e^{x^2}dx}~\text{and}~\int{\frac{\sin(x)}{x}dx}

can not be solved in terms of elementary functions. In fact, it is a criticism that I have for most calculus classes: they don't stress enough that most integrals are not solvable. When you come out of a calculus class, you think you can solve any integral, but this is simply false.

But numerical methods still work however...

 

[Metaleer]

While it's true that most integrals that one can think of have no elementary antiderivative, WolframAlpha claims that it can't find a result in terms of standard mathematical functions that it knows - and Wolfram Alpha knows many non-elementary functions like the Gaussian (erf and erfc) and sine integral that you gave as examples.

Numerical methods are usually left for numerical integration of definite integrals - remember that this is an indefinite integral that we're looking for, or primitive or antiderivative if you prefer (although a Taylor series expansion could be found, I guess, atleast for a limited interval).

What surprised me is that it appeared as a puzzle in a mathematical journal. Perhaps it's a typo? If it helps, I found it http://books.google.co.uk/books?id=KX6D6hefyA0C&printsec=frontcover", on page 84, MATYC 125.

 

[micromass]

Hmm, that is weird. I've checked some other mathematical software, but they also say that they cannot find a solution. So I think there's a typo somewhere. I have the feeling that they forgot to type the range of the integral, and that it's actually a definite integral...

 

[Metaleer]

Thanks for all the input, micromass. It may be possible to locate the original MATYC issue in question and see if there is indeed a typo and the integrand is messed up and/or it's actually a definite integral.