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Question Number 204909 by mnjuly1970 last updated on 01/Mar/24

$$$$$$\mathrm{\Omega }={\int }_{\frac{1}{e}}^e\frac{arctan\left(x\right)}{x}dx=?$$

Answered by witcher3 last updated on 01/Mar/24

$$\mathrm{x}\to \frac{1}{\mathrm{x}}$$$$\mathrm{\Omega }={\int }_{\frac{1}{\mathrm{e}}}^{\mathrm{e}}\frac{{\tan }^{-1}\left(\frac{1}{\mathrm{x}}\right)}{\frac{1}{\mathrm{x}}}.\frac{1}{{\mathrm{x}}^2}={\int }_{\frac{1}{\mathrm{e}}}^{\mathrm{e}}\frac{\frac{\pi }{2}-{\tan }^{-1}\left(\mathrm{x}\right)}{\mathrm{x}}=\frac{\pi }{2}\ln \left({\mathrm{e}}^2\right)-\mathrm{\Omega }$$$$\mathrm{\Omega }=\frac{\pi }{2}$$

Commented by mnjuly1970 last updated on 01/Mar/24

$$\cancel{⋚}$$

Commented by witcher3 last updated on 01/Mar/24

$$\mathrm{withe}\mathrm{Pleasur}\mathrm{Sir}$$

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