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Question Number 162604 by amin96 last updated on 30/Dec/21

$$I={\int }_0^{\frac{\pi }{4}}\mathit{x}\mathit{t}\mathit{g}\left(\mathit{x}\right)\mathit{d}\mathit{x}=?$$

Answered by Ar Brandon last updated on 30/Dec/21

$$I={\int }_0^{\frac{\pi }{4}}x\tan xdx$$$$=-{\left[x\ln \left(\cos x\right)\right]}_0^{\frac{\pi }{4}}+{\int }_0^{\frac{\pi }{4}}\ln \left(\cos x\right)dx$$$$=-\frac{\pi }{4}\ln \left(\frac{1}{\sqrt{2}}\right)+\frac{G}{2}-\frac{\pi \ln 2}{4}=\frac{G}{2}-\frac{\pi \ln 2}{8}$$

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