How Do You Integrate x^x dx?

This integral, sometimes called the “Bernoulli integral” after Johann Bernoulli who figured out in 1697 how to solve the related integral:

\int_0^1 \frac{1}{x^x} dx

In the form:

\int_0^x t^{at} dt

it has been called the “sophomore’s dream”:


How does one go from the “x” variables to the “t” variables? My limited mathematical understanding is that one cannot include on one side of the equation what is not on the other side of the equal sign. For instance x^2+2x+1=(x+1)(x+1).

I suppose you math guys are probably so advanced y’all have forgotten the FOIL method, ha!

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It’s just how you choose to write the equation. The variable over which you’re integrating is just an arbitrary place-holder. In the first case it’s “x” while in the second “t” was used. The second case is more general in that the upper limit of the integration is not fixed at 1 but rather can be any value, expressed here (confusingly) as x. There is the further generality that the exponent can be multiplied by any factor a, while in the original case a=1.