Saw this one on the univ's notice board...
I don't know how hard this will be for you people, especially Krizalid, but I'm interested to see the answer.
because there is no elementary function whose derivative is x^x.
I reckon someone should define one. That is why it is not easy by elementary means.
I did run it through the calculator and got a huge answer.
Maple nor the Integrator will give an indefinite integral for this. They just spit back the same thing.
Just for fun, I generated a graph with Maple using 1 to 10. For 1..100, the y value was too large.
Look how huge it is just using 1 to 10, let alone 1 to 100.
Afterall, this is quite the exponential. 2^2, 3^3, ....., 100^100.
But I still feel there must be an easier way of solving it. You know when you just have a feeling that there is a better, more elegant method?
Hacker got a typo, it should be or This identity is called as "Sophomore's Dream." One can prove it easily with galactus' trick (post #7) after series expansion and integration.
As for the problem, I dunno, I think I would've thought in numerical methods to tackle it too.
One trouble with Taylor expansion is where to center it at... 1? 50? 100?
The derivatives of the function are rather beautiful...
I think you get the idea...
Let's choose for simplicity...
This is obviously not enough of an expansion for . More work needs to be done, but this is a start.
Johann Bernoulli first calculated in a paper he published in 1697. The series converges extremely rapidly (an attribute noted by Bernoulli) .... it only takes a few terms to calculate the integral accurately to ten decimal places.
Krizalid is quite right in what he says regarding the proof:
and each of these integrals is simple to do. Especially if you know the reduction formula