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Question Number 197783 by pticantor last updated on 28/Sep/23

$$\int \frac{x.\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{g}\left(\mathit{x}\right)}{{\mathit{x}}^2+1}\mathit{d}\mathit{x}=?$$$$$$$$$$$$$$$$$$

Answered by EmGent last updated on 29/Sep/23

$$\frac{1}{2}{arctg}^2\left(x\right)+c$$

Commented by mr W last updated on 29/Sep/23

$$but\frac{d}{dx}\left(\frac{1}{2}{arctg}^2\left(x\right)\right)=\frac{arctg\left(x\right)}{x^2+1}\ne \frac{xarctg\left(x\right)}{x^2+1}$$

Commented by pticantor last updated on 29/Sep/23

$$infact$$

Answered by witcher3 last updated on 30/Sep/23

$$\mathrm{x}=\mathrm{tg}\left(\mathrm{y}\right),$$$$\iff \int \mathrm{ytg}\left(\mathrm{y}\right)\mathrm{dy}\mathrm{IBP}$$$$=-\mathrm{yln}(\cos \left(\mathrm{y}\right)+\int \ln \left(\cos \left(\mathrm{y}\right)\right)\mathrm{dy}$$$$\int \ln \left(\cos \left(\mathrm{y}\right)\right)\mathrm{dy}=\int \ln \left(\frac{{\mathrm{e}}^{\mathrm{iy}}+{\mathrm{e}}^{-\mathrm{iy}}}{2}\right)\mathrm{dy}$$$$=\mathrm{z}={\mathrm{e}}^{\mathrm{iy}}\Rightarrow \mathrm{dz}={\mathrm{ie}}^{\mathrm{iy}}\mathrm{dy},\mathrm{dy}=\frac{\mathrm{dz}}{\mathrm{iz}}$$$$\int \ln \left(\cos \left(\mathrm{y}\right)\right)\mathrm{dy}=\int \ln \left(\frac{\mathrm{z}+\frac{1}{\mathrm{z}}}{2}\right)\frac{\mathrm{dz}}{\mathrm{iz}}$$$$=\int \frac{\ln ({\mathrm{z}}^2+1)-\ln \left(\mathrm{z}\right)-\ln \left(2\right)}{\mathrm{iz}}\mathrm{dz}$$$$=\mathrm{iln}\left(2\right)\ln \left(\mathrm{z}\right)+\frac{\mathrm{i}}{2}{\ln }^2\left(\mathrm{z}\right)-\mathrm{i}\int \frac{\ln (1+{\mathrm{z}}^2)}{\mathrm{z}}\mathrm{dz}$$$$\int \frac{\ln (1+{\mathrm{z}}^2)}{\mathrm{z}}\mathrm{dz}=\int \frac{\ln (1+{\mathrm{z}}^2)}{{\mathrm{z}}^2}\mathrm{zdz}=\frac{1}{2}\int \frac{\ln (1+\mathrm{t})}{\mathrm{t}}\mathrm{dt}$$$$=\frac{1}{2}\int \frac{\ln (1-(-\mathrm{t}))}{(-\mathrm{t})}\mathrm{d}(-\mathrm{t})$$$$=-\frac{1}{2}{\mathrm{Li}}_2(-\mathrm{t})$$$$\int \frac{\ln (1+{\mathrm{z}}^2)}{\mathrm{z}}=-\frac{1}{2}{\mathrm{Li}}_2(-{\mathrm{z}}^2)$$$$\int \ln \left(\cos \left(\mathrm{y}\right)\right)\mathrm{dy}$$$$-\ln \left(2\right)\mathrm{y}-\mathrm{i}\frac{{\mathrm{y}}^2}{2}+\frac{\mathrm{i}}{2}{\mathrm{Li}}_2(-{\mathrm{e}}^{2\mathrm{iy}})$$$$\int \frac{{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right)}{1+{\mathrm{x}}^2}\mathrm{dx}=\mathrm{xln}\left(\cos \left(\mathrm{x}\right)\right)-\ln \left(2\right){\tan }^{-1}\left(\mathrm{x}\right)-\frac{\mathrm{i}}{2}{\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2$$$$+\frac{\mathrm{i}}{2}{\mathrm{Li}}_2(-{\mathrm{e}}^{{2\mathrm{itan}}^{-1}\left(\mathrm{x}\right)})+\mathrm{c}$$$$$$$$$$$$$$

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