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I'm interested in knowing if the indefinite integral of $x^x$ can be found in terms of elementary functions.

I am under the impression (be it correct or incorrect) that it can be found. This is why: the derivative of $x^x$ has $x^x$ in it ($d/dx[x^x] = x^x(\ln(x) + 1)$). The derivative is quite easy to find with logarithmic differentiation, but the integral – not so much.

If the indefinite integral cannot be defined in terms of elementary functions, why?

If the indefinite integral can be found, would you please work it out? I find myself lost here, I've tried multiple methods.

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Go to wolframalpha.com, enter something like int(x^x,x) into the box, and see what happens. I would guess that if it gives any answer it will be in terms of special functions (not elementary functions). –  Stefan Smith Apr 29 '13 at 19:15
    
It can't see here. –  Raymond Manzoni Apr 29 '13 at 22:05
    
Let $F(x) = \int x^x\,\mathrm dx$. Then $F'(x)=x^x$. Let's make the ansatz $F(x) = x^x f(x)$. Then $F'(x) = x^x((1+\ln x)f(x) + f'(x))$. Therefore $f(x)$ should solve the differential equation $f'(x) + (1+\ln x)f(x) = 1$. Unfortunately I have no idea how you would solve that differential equation. –  celtschk Aug 1 '13 at 14:08
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This function has no elementary antiderivative. Of course, it can be integrated numerically over given intervals (using a series expansion or other methods). This is a famous example. See here. –  Andres Caicedo Sep 1 '13 at 19:28
    
There are a few questions on this site listing references on this topic, for example this one. –  Andres Caicedo Sep 1 '13 at 19:30

3 Answers 3

Hint: $x^x=e^{x \ln x}=\sum_{k=0}^{\infty}\frac{(x \ln x)^k}{k!}$. Then interchange the integral sign and summation sign.

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maybe you will find it useful: en.wikipedia.org/wiki/Sophomore's_dream –  Ishigami Apr 29 '13 at 19:22
    
I've tried integrating e^xln(x) multiple times, but I haven't gotten very far. The summation method may be useful. –  Thomas Kent Apr 29 '13 at 19:25

I remember doing a STEP question, which basically lead to solving this integral http://www.mathshelper.co.uk/STEP%20III%202009.pdf q8.

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No, $\int x^x\;dx$ is not an elementary function.

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