How can the following integral be calculated:
In=∫10∫10⋯∫10∏nk=1(1−xk1+xk)1−∏nk=1xkdx1⋯dxn−1dxn
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=∫10∫10⋯∫10∏nk=1(1−xk1+xk)1+∏nk=1xkdx1⋯dxn−1dxn
Again, it can be shown easily, that J1=1−ln(2).
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