- #1

alyafey22

Gold Member

MHB

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$$I(n,m) = \int^1_0 \log^n(x)\log^m(1-x)\,dx$$

Our purpose is finding a closed form for the general case.

Note: for a given n and m the above formula can be deduced by succesive differentiation of the beta representation

$$B(p,q) = \int^1_0 x^{p-1} (1-x)^{1-q}\,dx$$

Yet , the computations are very complicated. The main goal is tackling the question using different approaches , possibly better.

This is

**NOT**a tutorial , any suggestions or attempts are always welcomed.