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Question Number 205264 by pticantor last updated on 13/Mar/24

$$\mathit{p}\mathit{l}\mathit{s}\mathit{h}\mathit{o}\mathit{w}\mathit{t}\mathit{o}\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{t}\mathit{h}\mathit{i}\mathit{s}?$$$${\int }_{\frac{1}{2}}^1\frac{\mathit{l}\mathit{n}(\mathit{x}+1)}{\mathit{x}}\mathit{d}\mathit{x}$$

Answered by Berbere last updated on 13/Mar/24

$$=-(-{\int }_{\frac{1}{2}}^1\frac{ln(1-(-x))}{(-x)}d(-x))$$$$=-{\left[{Li}_2(-x)\right]}_{\frac{1}{2}}^1=-{Li}_2(-1)+{Li}_2(-\frac{1}{2})$$$$=\frac{{\pi }^2}{12}+{Li}_2(-\frac{1}{2})$$

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