$\int x^x\,dx$ - What is it, and why?

Question

As a high school calculus student, I stumbled across the possibilities for: $$\int x^x\,dx$$ My friends and I are currently stumped. My first idea was: $$ \left(\frac{1}{x+1}\right) x^{x+1} $$ But I doubt it is this simple. I've read through multiple forums, and cannot find any solution or explanation! Please help shed some light on this mysterious intregral.

Answer 1

$x^x$ cannot be integrated since we have found no function whose derivative is $x^x$, though we can find approximations of definite integrals of $x^x$ by considering it's power series.

$$x^x=1+x\ln(x)+\frac{1}{2}x^2\ln^2(x)+\frac{1}{6}x^3\ln^3(x)+...=\sum_{n=0}^\infty \frac{x^n\ln^n(x)}{n!}$$

Which therefore means that

$$\int_a^b x^x dx=\int_a^b \left(\sum_{n=0}^\infty \frac{x^n\ln^n(x)}{n!}\right)dx$$

Although this certainly seems a whole lot messier, we can now approximate values of the impossible function $\displaystyle f(s)=\int_0^s x^x dx$

Answer 2

To elaborate a bit on my comment, it turns out that $f(x) = x^x$ has no anti-derivative expressible in terms of elementary functions.

The math involved is at a university level. If I recall correctly, it is a decidable problem to check if a given function has an anti-derivative expressible in terms of a given class of elementary functions.

The result can either be derived from something called differential Galois theory or from Liouville theory, where the latter is probably the easier field to get into of the two, but is still somewhat high-level and technical (in fact, the two fields are probably related).

This is just to tell you that it is possible to show that a given function has no "nice" integral, but also to tell you that doing so is advanced.

Answer 3

If we assume the antiderivative of $x^x$ is in the form of $x^xf(x)$, then we can see that $f(x)$ is the solution to the differential equation $$1=(\ln(x)+1)f(x)+f'(x)$$

Just a thought.

Answer 4

As diligar answered, if you write $$x^x=\sum_{n=0}^\infty \frac{x^n\ln^n(x)}{n!}$$ you are then let with the problem of computing the integrals $$I_n=\int x^n\log^n(x)\,dx$$ which is doable using the gamma function. It simplifies using the exponential integral function and the result is $$I_n=-\log ^{n+1}(x)\, E_{-n}\big(-(n+1) \log (x)\big)$$ The problem is that the convergence requires quite many terms. To give you an idea, using the above expressions, I computed using numerical integration the value $$\int_2^9 x^x\,dx \approx 1.22592630615242\times 10^8$$ Now, using the summation for $k$ terms $$\left( \begin{array}{cc} k & value \\ 10 & 2.95976367580423\times 10^6 \\ 15 & 2.90442516168767\times 10^7 \\ 20 & 8.13458442990505\times 10^7 \\ 25 & 1.13993643109235\times 10^8 \\ 30 & 1.21744554000771\times 10^8 \\ 35 & 1.22550640678100\times 10^8 \\ 40 & 1.22591514498982\times 10^8 \\ 45 & 1.22592613687818\times 10^8 \\ 50 & 1.22592630460872\times 10^8 \\ 55 & 1.22592630614356\times 10^8 \\ 60 & 1.22592630615238\times 10^8 \\ 65 & 1.22592630615241\times 10^8 \\ 70 & 1.22592630615241\times 10^8 \end{array} \right)$$

Answer 5

You can indeed easily integrate

$$\int x^a dx=\frac{x^{a+1}}{a+1}$$and

$$\int a^xdx=\frac{a^x}{\ln(a)}.$$

Check by taking the derivatives.

But

$$\int x^x dx$$ is a completely different matter, and actually you can't express it using the usual functions.

In the previous examples, the exponentiation is somewhat conserved so we would expect an antiderivative like $x^x$. But taking the derivative, a nasty factor appears.

$$\left(x^x\right)'=(1+\ln(x))x^x.$$

You can try a more general form like $f(x)^{g(x)}$, and that yields

$$\left(f(x)^{g(x)}\right)'=\left(\frac{f'(x)}{f(x)}g(x)+\ln(f(x))g'(x)\right)f(x)^{g(x)}$$ but there is little hope for simplification.

Answer 6

Plain and simple: with differentiation and integration, the most important thing is what's the variable of integration/differentiation. Your integrand $x^x$ changes both the exponent and the base when you vary $x$, so it has nothing to do with functions of the form $a^x$ or $x^a$.

This integral isn't expressible with what we know as elementary functions (powers,log,trig,exp), but that's not surprising. Most functions are that way: only a very narrow set of functions can actually be integrated in a closed form, and even that is usually very hard (a lot of instinct, tricks and guesswork goes into computing integrals analytically - it's quite different from differentiation, which is a routine process a quick recipe can do).

Some more significant and useful integrals (and infinite sums, and solutions of transcendental equations,...) are conveniently defined as special functions, reliable algorithms are designed, tables are made, and routines in programming libraries are created. Examples are elliptic functions, gamma function, Fresnel integrals, Lambert function, Riemann zeta function,... if for whatever reason, your integral was important and frequently seen in mathematics, someone would name it. But for most expressions that aren't integrable within elementary functions, you see them once and never again. So we just integrate them numerically with one of the many reliable methods; in a sense, it's no different than what special functions from math libraries are doing, or what the calculator does when you press sin. It's just different in our heads, because we're so used to some functions having a special name and the rules that apply to them, we're uncomfortable when something new comes along.