Wednesday, March 2, 2016

sequences and series - How to evaluate int10fraclog2(1+x)xmathrmdx?

The definite integral




10log2(1+x)xdx=ζ(3)4



arose in my answer to this question. I couldn't find it treated anywhere online. I eventually found two ways to evaluate the integral, and I'm posting them as answers, but they both seem like a complicated detour for a simple result, so I'm posting this question not only to record my answers but also to ask whether there's a more elegant derivation of the result.



Note that either using the method described in this blog post or substituting the power series for log(1+x) and using



1k1sk=1s(1k+1sk)



yields




10log2(1+x)xdx=2n=1(1)n+1Hn(n+1)2.



However, since the corresponding identity without the alternating sign is used to obtain the sum by evaluating the integral and not vice versa, I'm not sure that this constitutes progress.

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