Tuesday, 5 September 2017

calculus - How to compute the following integral in n variables?

How can the following integral be calculated:
In=101010nk=1(1xk1+xk)1nk=1xkdx1dxn1dxn
There should be n integral signs, but I didn't know how to write that.



It is easy to show that I1=ln(2). After partial fractioning and the help of Wolfram Alpha, I managed to show that I2=4ln(2)2ln2(2)π26.



But how to derive a general result? Any help would be highly appreciated!



Edit:




As a supplementary question, how to calculate this slightly modified integral:
Jn=101010nk=1(1xk1+xk)1+nk=1xkdx1dxn1dxn
Again, it can be shown easily, that J1=1ln(2).

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