Integrating $x^x$ and getting a graph

Question

I've heard many times of functions that cannot be integrated. For example, $x^x$, which is the most common. But what I don't know is how could you, even if the graph has no equation, plot this integral. If anyone could give me a graph, I would be extremely pleased. Or at least, could you tell me how to graph it? Maybe in Wolfram Mathematica?

Answer 1

Here is a quick picture of $\ x\mapsto \int_0^x t^t\;dt$

I added the plot of the real and imaginary part of the integral in the complex plane $\ z\mapsto \int_0^z t^t\;dt\ $ with $z=x+iy$ (note the 'branch jump' on the negative axis $(-\infty,0)$).

Answer 2

Here is the graph of the function with Maple

Plot( $x^x$, $x=0..2$ );

The graph of the integral of $x^x$ using the truncated Taylor series

$$ x^x=1+\ln \left( x \right) x+\frac{1}{2}\, \left( \ln \left( x \right) \right) ^{2}{x}^{2}+\frac{1}{6}\, \left( \ln \left( x \right) \right) ^{3}{x}^{3}+\frac{1}{ 24}\, \left( \ln \left( x \right) \right) ^{4}{x}^{4}+O(x^5). $$

Answer 3

Here is an interesting thread for you: http://www.physicsforums.com/showthread.php?t=77605

Quote: The integral of x^x between 0 and 1 was calculated by Johann Bernoulli in 1697 using power series (not Euler). The proof appears in "Opera Omnia" vol. 3 (1697) pp. 376 - 381.