Question Number 218626 by Nicholas666 last updated on 13/Apr/25 $$ \\ $$$$\:{Prove}\:{that}\:{for}\:{all}\:{real}\:{numbers}\:{a}\:{and}\:{b} \\ $$$${with}\:{a}<{b},\:{the}\:{following}\:{inequality}\:{holds}; \\ $$$$\left(\int_{{a}} ^{{b}} \mathrm{1}\:{dx}\right)^{\mathrm{3}} \leqslant\:\left({b}−{a}\right)\left(\int_{{a}} ^{{b}} \left({x}−{a}+\mathrm{1}\right)^{\mathrm{2}} {dx}\right)\left(\int_{{a}\:} ^{{b}} \frac{\mathrm{1}}{\left({a}−{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}\right)…
Question Number 218583 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\int_{\mathrm{0}\:} ^{\infty} \:{e}^{−{x}} \:\left(\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{f}\left({n}\right)}{{n}}\:{sin}\left({nx}\right)\right){dx}\:=\mathrm{1} \\ $$$$ \\ $$ Answered by aleks041103 last…
Question Number 218582 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({ax}\right)\:−\:{cos}\left({bx}\right)}{{x}^{\mathrm{2}} }\:{dx}\:=\:\frac{\pi}{\mathrm{2}}\:\mid{b}−{a}\mid\: \\ $$$$ \\ $$ Answered by aleks041103 last updated on…
Question Number 218599 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\infty} \boldsymbol{{x}}^{\boldsymbol{{s}}−\mathrm{1}} \:\underset{\boldsymbol{{n}}=\mathrm{1}\:} {\overset{\infty} {\prod}}\left(\mathrm{1}−\boldsymbol{{e}}^{−\boldsymbol{{nx}}} \right)^{−\mathrm{24}} \:\boldsymbol{{dx}} \\ $$$$ \\ $$ Commented by…
Question Number 218598 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\:\:\underset{\mathrm{0}} {\int}^{\infty} \:\frac{\boldsymbol{{x}}}{\boldsymbol{{sinh}}\left(\boldsymbol{{x}}\right)}\boldsymbol{{ln}}\left(\frac{\boldsymbol{{x}}}{\boldsymbol{{e}}^{\boldsymbol{{x}}} −\mathrm{1}}\right)\boldsymbol{{dx}}\: \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 218560 by MrGaster last updated on 12/Apr/25 Commented by MrGaster last updated on 12/Apr/25 $${J}_{\mathrm{0}} \left({a}\sqrt{\mathrm{1}−{u}^{\mathrm{2}} }\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {a}^{\mathrm{2}{n}} }{\left({n}!\right)^{\mathrm{2}} \mathrm{2}^{\mathrm{2}{n}} }\left(\mathrm{1}−{u}^{\mathrm{2}}…
Question Number 218563 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:{solve}\:{for}\:\boldsymbol{{x}}\:\in\:\mathbb{R}\: \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}\left({xt}\right)}{{e}^{{t}} −\mathrm{1}\:}\:{dt}\:=\:\frac{\boldsymbol{\pi}}{\mathrm{2}}\:{coth}\left(\boldsymbol{\pi}{x}\right)\:−\:\frac{\mathrm{1}}{\mathrm{2}{x}}\:\: \\ $$$$ \\ $$ Terms of Service Privacy…
Question Number 218594 by MrGaster last updated on 12/Apr/25 $$\mathrm{A}\:\mathrm{kind}\:\mathrm{of}\:\mathrm{calculation}\:\mathrm{relatedt} \\ $$$$\mathrm{o}\:\mathrm{arctangent}\:\mathrm{integral}: \\ $$$$\boldsymbol{\mathrm{Exere}}:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{arctan}^{\mathrm{2}} }{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}. \\ $$$$\mathrm{Solution}:\underset{{A}} {\underbrace{=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{arctan}^{\mathrm{2}} {x}}{{x}}{dx}}}\:−\underset{{B}} {\underbrace{\int_{\mathrm{0}}…
Question Number 218462 by York12 last updated on 10/Apr/25 $$\int\frac{{dx}}{\:\sqrt{\mathrm{2}{x}−{x}^{\mathrm{2}} +\mathrm{3}}} \\ $$ Answered by som(math1967) last updated on 10/Apr/25 $$\int\frac{{dx}}{\:\sqrt{\mathrm{4}−\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1}\right)}} \\ $$$$=\int\frac{{dx}}{\:\sqrt{\mathrm{2}^{\mathrm{2}} −\left({x}−\mathrm{1}\right)^{\mathrm{2}}…
Question Number 218398 by Nicholas666 last updated on 09/Apr/25 Answered by MrGaster last updated on 10/Apr/25 Terms of Service Privacy Policy Contact: info@tinkutara.com