How to calculate \int_0^1 x^xdx using series? I read from

Nylah Church

Nylah Church

Answered question

2022-01-23

How to calculate 01xxdx using series? I read from a book that
01xxdx=1122+133++(1)n1(n+1)n+1+

Answer & Explanation

Palandriy0

Palandriy0

Beginner2022-01-24Added 14 answers

Just write
xx=exlnx=n=0(xlnx)nn!
and use that
01(xlnx)ndx=(1)n0yne(n+1)ydy,
To show the last formula, make the change of variables x=ey so that
01(xlnx)ndx=(1)n0yne(n+1)ydu,
which is clearly expressible in terms of the Gamma function.

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