Question Number 122626 by rs4089 last updated on 18/Nov/20 Answered by Dwaipayan Shikari last updated on 18/Nov/20 $${S}=\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+…=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${But}\:{actually}\:{sum}\:{does}\:{not}\:{exist} \\ $$$$\zeta\left(\mathrm{1}−{s}\right)=\mathrm{2}^{\mathrm{1}−{s}} \pi^{−{s}} {cos}\left(\frac{\pi{s}}{\mathrm{2}}\right)\Gamma\left({s}\right)\zeta\left({s}\right) \\…
Question Number 122625 by mnjuly1970 last updated on 18/Nov/20 $$\:\:\:\:\:\:\:\:\:…\:{advanced}\:\:{integral}… \\ $$$$\:\:\:\:\:{i}:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{\mathrm{1}}{{ln}\left({x}\right)}+\frac{\mathrm{1}}{\mathrm{1}−{x}}\right){dx}=\gamma \\ $$$$\:\:\:\:\:\:{ii}:\:\psi\left({x}\right)=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{{e}^{−{t}} }{{t}}\:−\frac{{e}^{−{tx}} }{\mathrm{1}−{e}^{−{t}} }\right){dt} \\ $$$$\:\:\:\:{solution}\::\left\{_{\mathrm{2}\::\:{ln}\left({n}\right)\:\overset{{easy}} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}}…
Question Number 122614 by john santu last updated on 18/Nov/20 Answered by liberty last updated on 18/Nov/20 $$\:{F}=\int\:\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)} \\ $$$${By}\:{trigonometry}\:{substitution}\:{method} \\ $$$${let}\:{x}=\:\mathrm{tan}\:\theta\: \\…
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Question Number 122588 by bramlexs22 last updated on 18/Nov/20 $${Find}\:{the}\:\mathrm{arc}\:{length}\:{of}\:{the}\:{curve}\: \\ $$$${x}=\frac{\mathrm{1}}{\mathrm{4}}{y}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\left({y}\right)\:{between}\:{the}\:{points} \\ $$$${with}\:{the}\:{ordinates}\:{y}=\mathrm{1}\:{and}\:{y}=\mathrm{2}. \\ $$$$ \\ $$ Answered by MJS_new last updated on…
Question Number 122587 by bramlexs22 last updated on 18/Nov/20 $$\:\:\underset{\mathrm{1}} {\overset{\mathrm{16}} {\int}}\:\mathrm{arctan}\:\sqrt{\sqrt{{x}}\:−\mathrm{1}}\:{dx}\: \\ $$$$\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\mathrm{sin}\:\left(\mathrm{2}{x}\right)\:\mathrm{arctan}\:\left(\mathrm{sin}\:{x}\right)\:{dx}\: \\ $$ Answered by MJS_new last updated on 18/Nov/20…
Question Number 122572 by rs4089 last updated on 18/Nov/20 Answered by Olaf last updated on 18/Nov/20 $$\mathrm{With}\:\mathrm{Euler}'\mathrm{s}\:\mathrm{Gamma}\:\mathrm{function}\:\Gamma\:: \\ $$$$\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} \mathrm{ln}{xdx}\:=\:\Gamma'\left(\mathrm{1}\right)\:=\:−\gamma \\ $$$$\mathrm{where}\:\gamma\:\mathrm{is}\:\mathrm{the}\:\mathrm{Euler}−\mathrm{Mascheroni} \\…
Question Number 188086 by universe last updated on 25/Feb/23 $$\:\:\:\: \\ $$$$\:\:\:\:\:\:\mathrm{I}\:\:\:=\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{tan}^{−\mathrm{1}} \left({x}/{a}\right)}{{x}\left({x}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)}{dx} \\ $$ Answered by witcher3 last updated on…
Question Number 122544 by bramlexs22 last updated on 18/Nov/20 $$\:{any}\:{one}\:{can}\:{explain}\:{me}\:{about} \\ $$$${Fresnel}\:{integral}\:? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 122542 by bramlexs22 last updated on 18/Nov/20 $$\:\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:\sqrt{\mathrm{1}+\mathrm{cos}\:\left({x}^{\mathrm{2}} \right)}\:{dx}\:? \\ $$ Answered by Olaf last updated on 18/Nov/20 $$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{\mathrm{1}+\mathrm{cos}\left({x}^{\mathrm{2}}…