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Category: Integration

prove-that-0-ln-1-x-j-0-x-dx-ln-2-Hint-1-j-0-x-n-0-1-n-x-2n-2-2n-2-n-1-Bessel-function-Hint-2-L-j-0-x-

Question Number 142362 by mnjuly1970 last updated on 30/May/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\:{prove}\:\:{that}: \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {ln}\left(\frac{\mathrm{1}}{{x}}\right).{j}_{\mathrm{0}} \left({x}\right){dx}:=\:\gamma+{ln}\left(\mathrm{2}\right)\: \\ $$$$\:\:\:\:\:{Hint}:\left(\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:{j}_{\mathrm{0}} \left({x}\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}}…

Question-76826

Question Number 76826 by Master last updated on 30/Dec/19 Answered by john santu last updated on 31/Dec/19 $$=\:\int\underset{\mathrm{0}} {\overset{\mathrm{2}} {\:}}\:\int\underset{\mathrm{0}} {\overset{\:\mathrm{1}} {\:}}\:\int\underset{\mathrm{0}} {\overset{\:\mathrm{3}} {\:}}\:\left(\mathrm{2}{y}+{z}+{x}\right)\:{dz}\:{dy}\:{dx}\: \\…

e-1-e-1-ln-x-ln-ln-x-dx-

Question Number 142344 by mnjuly1970 last updated on 30/May/21 $$ \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}:=\underset{\frac{\mathrm{1}}{{e}}} {\int}^{\:{e}} \left\{\frac{\mathrm{1}}{{ln}\left({x}\right)}+{ln}\left({ln}\left({x}\right)\right)\right\}{dx} \\ $$ Commented by Dwaipayan Shikari last updated on 30/May/21 $${log}\left({x}\right)={t}…

0-sinx-x-dx-

Question Number 142338 by rs4089 last updated on 30/May/21 $$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}}{{x}^{\mu} }{dx}\:\:=?\:\:\: \\ $$ Answered by Dwaipayan Shikari last updated on 30/May/21 $$\frac{\mathrm{1}}{{x}^{\mu} }=\frac{\mathrm{1}}{\Gamma\left(\mu\right)}\int_{\mathrm{0}}…