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Question Number 201011 by Calculusboy last updated on 28/Nov/23

$$\mathit{P}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{a}\mathit{t}$$$${\int }_0^{\mathrm{\infty }}\frac{2\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{a}\mathit{n}\left(\frac{\mathit{t}}{\mathit{x}}\right)}{{\mathit{e}}^{2\mathit{\pi }\mathit{t}}-1}\mathit{d}\mathit{t}=\mathit{I}\mathit{n}\mathbf{\Gamma }\left(\mathit{x}\right)-\mathit{x}\mathit{I}\mathit{n}\left(\mathit{x}\right)+\mathit{x}-\frac{1}{2}\mathit{I}\mathit{n}\left(\frac{2\mathit{\pi }}{\mathit{x}}\right)$$$$\mathit{M}\mathit{i}\mathit{c}\mathit{h}\mathit{a}\mathit{e}\mathit{l}\mathit{f}\mathit{a}\mathit{r}\mathit{a}\mathit{d}\mathit{a}\mathit{y}$$

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