What has Industrial Design got to do with mathematics anyway?
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The integral of x^x dx. I like a good puzzle,...sets my mind working.
Ok...x^x needs to be expressed as a Power Series. As I mentioned above,..express x^x as e^xlnx and then using the Maclaurin Series for e^x expand e^xlnx. In closed form this is the summation from n = 0 to infinity of (x^n(lnx)^n)/n!. This expression can be integrated term by term and there is a considerable quantity of work to cover in order to obtain the solution.
Limits are set from 0 to 1 and the integral of x^x dx is finally shown to be the summation from n = 1 to infinity of (-1)^n-1/n^n
Ron.