Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 136036 by mathmax by abdo last updated on 18/Mar/21

$$\mathrm{calculate}{\int }_0^{\mathrm{\infty }}\frac{\mathrm{logx}}{{\mathrm{x}}^4+{\mathrm{x}}^2+1}\mathrm{dx}$$

Commented by Ajetunmobi last updated on 18/Mar/21

$$$$$$\mathbf{hi}\mathbf{sir}\mathbf{how}\mathbf{can}\mathbf{we}\mathbf{chat}\mathbf{privately}?$$

Commented by Ajetunmobi last updated on 18/Mar/21

Commented by liki last updated on 19/Mar/21

$$\mathrm{Drop}\mathrm{your}\mathrm{whatssap}\mathrm{number}\mathrm{here}!$$

Commented by mathmax by abdo last updated on 31/May/21

$$\mathrm{you}\mathrm{are}\mathrm{right}\mathrm{liki}...$$

Answered by Ar Brandon last updated on 18/Mar/21

$$\mathrm{\Delta }={\int }_0^{\mathrm{\infty }}\frac{\mathrm{lnx}}{{\mathrm{x}}^4+{\mathrm{x}}^2+1}\mathrm{dx}$$$$={\int }_0^1\frac{\mathrm{lnx}}{{\mathrm{x}}^4+{\mathrm{x}}^2+1}\mathrm{dx}+{\int }_1^{\mathrm{\infty }}\frac{\mathrm{lnx}}{{\mathrm{x}}^4+{\mathrm{x}}^2+1}\mathrm{dx}$$$$={\int }_0^1\frac{\mathrm{lnx}}{{\mathrm{x}}^4+{\mathrm{x}}^2+1}\mathrm{dx}-{\int }_0^1\frac{{\mathrm{x}}^2\mathrm{lnx}}{{\mathrm{x}}^4+{\mathrm{x}}^2+1}\mathrm{dx}$$$$={\int }_0^1\frac{(1-{\mathrm{x}}^2)\mathrm{lnx}}{(1-{\mathrm{x}}^6)}\mathrm{dx}-{\int }_0^1\frac{{\mathrm{x}}^2(1-{\mathrm{x}}^2)\mathrm{lnx}}{(1-{\mathrm{x}}^6)}\mathrm{dx}$$$$=\frac{1}{36}{\int }_0^1\frac{({\mathrm{u}}^{-\frac{5}{6}}-{\mathrm{u}}^{-\frac{1}{2}})\mathrm{lnu}}{1-\mathrm{u}}\mathrm{du}-\frac{1}{36}{\int }_0^1\frac{({\mathrm{u}}^{-\frac{1}{2}}-{\mathrm{u}}^{-\frac{1}{6}})\mathrm{lnu}}{1-\mathrm{u}}\mathrm{du}$$$$=-\frac{1}{36}\{{\psi }^1\left(\frac{1}{6}\right)-{\psi }^1\left(\frac{1}{2}\right)-{\psi }^1\left(\frac{1}{2}\right)+{\psi }^1\left(\frac{5}{6}\right)\}$$$$=-\frac{1}{36}\{\frac{{\pi }^2}{{\sin }^2\left(\frac{\pi }{6}\right)}-2{\psi }^1\left(\frac{1}{2}\right)\}=-\frac{1}{36}\{4{\pi }^2-2\times 4\times \frac{3}{4}\zeta \left(2\right)\}$$$$=-\frac{(4{\pi }^2-{\pi }^2)}{36}=-\frac{{\pi }^2}{12}★$$

Commented by Dwaipayan Shikari last updated on 18/Mar/21

$$$$😃

Commented by Ajetunmobi last updated on 18/Mar/21

$$$$$$\mathbf{you}\mathbf{did}\mathbf{the}\mathbf{same}\mathbf{solution}\mathbf{as}\mathbf{mine}$$$$\mathbf{but}\mathbf{why}?$$

Commented by Ar Brandon last updated on 18/Mar/21

$$\mathrm{Because}\mathrm{I}\mathrm{wanna}\mathrm{learn}\mathrm{from}\mathrm{you}\mathrm{too}.$$$$\mathrm{Just}\mathrm{as}\mathrm{I}\mathrm{do}\mathrm{with}\mathrm{the}\mathrm{other}\mathrm{members}.$$😄😇

Commented by Ajetunmobi last updated on 18/Mar/21

$$$$$$\mathbf{ok}\mathbf{sir}\mathbf{we}\mathbf{are}\mathbf{all}\mathbf{learnig}\mathbf{from}\mathbf{each}$$$$\mathbf{other}$$

Commented by Ar Brandon last updated on 18/Mar/21

😀

Commented by SLVR last updated on 13/Apr/21

$$goodeveningmr.ar.Brandon$$$$kidlyhelpmewhatis{\psi }^{\prime }\left(1/6\right)..$$$$howitis$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com