Question and Answers Forum

All Questions      Topic List

Matrices and Determinants Questions

Previous in All Question      Next in All Question      

Previous in Matrices and Determinants      Next in Matrices and Determinants      

Question Number 198156 by Erico last updated on 12/Oct/23

$$\mathrm{Prove}\mathrm{that}$$$$\frac{2\mathrm{t}-1}{\mathrm{lnt}-\ln (1-\mathrm{t})}={\underset{0}{\int }}^1{\mathrm{t}}^{\mathrm{x}}{(1-\mathrm{t})}^{1-\mathrm{x}}\mathrm{dx}$$$$\mathrm{and}{\underset{0}{\int }}^1\frac{2\mathrm{t}-1}{\mathrm{lnt}-\ln (1-\mathrm{t})}\mathrm{dt}=\frac{\pi }{2}{\underset{0}{\int }}^1\frac{\mathrm{x}(1-\mathrm{x})}{\sin \left(\pi \mathrm{x}\right)}\mathrm{dx}$$

Answered by witcher3 last updated on 12/Oct/23

$${\int }_0^1{\mathrm{e}}^{(1-\mathrm{x})\ln (1-\mathrm{t})+\mathrm{xln}\left(\mathrm{t}\right)}\mathrm{dx}$$$$={\int }_0^1(1-\mathrm{t}){\mathrm{e}}^{\mathrm{x}(\ln \left(\frac{\mathrm{t}}{1-\mathrm{t}}\right)}\mathrm{dx}$$$$=\frac{(1-\mathrm{t})(\frac{\mathrm{t}}{1-\mathrm{t}}-1)}{\ln \left(\mathrm{t}\right)-\ln (1-\mathrm{t})}=\frac{2\mathrm{t}-1}{\ln \left(\mathrm{t}\right)-\ln (1-\mathrm{t})}$$$${\int }_0^1\frac{2\mathrm{t}-1}{\ln \left(\mathrm{t}\right)-\ln (1-\mathrm{t})}\mathrm{dt}={\int }_0^1{\int }_0^1{\mathrm{t}}^{\mathrm{x}}{(1-\mathrm{t})}^{1-\mathrm{x}}\mathrm{dxdt}$$$$={\int }_0^1{\int }_0^1{\mathrm{t}}^{\mathrm{x}}{(1-\mathrm{t})}^{1-\mathrm{x}}\mathrm{dxdt}={\int }_0^1{\int }_0^1{\mathrm{t}}^{\mathrm{x}}{(1-\mathrm{t})}^{1-\mathrm{x}}\mathrm{dtdx}$$$$={\int }_0^1\beta (\mathrm{x}+1,2-\mathrm{x})\mathrm{dx}$$$$={\int }_0^1\frac{\mathrm{x}(1-\mathrm{x})}{\mathrm{\Gamma }\left(3\right)}.\mathrm{\Gamma }\left(\mathrm{x}\right)\mathrm{\Gamma }(1-\mathrm{x})\mathrm{dx}=\frac{1}{2}{\int }_0^1\frac{\mathrm{x}(1-\mathrm{x})\pi }{\sin \left(\pi \mathrm{x}\right)}$$$$=\frac{\pi }{2}{\int }_0^1\frac{\mathrm{x}(1-\mathrm{x})}{\sin \left(\pi \mathrm{x}\right)}\mathrm{dx}=\frac{\pi }{2}.7\frac{\zeta \left(3\right)}{{\pi }^3}=\frac{7\zeta \left(3\right)}{2{\pi }^2}$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com