Question Number 188192 by universe last updated on 26/Feb/23 $$\:\:\: \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{a}^{\mathrm{2}} {x}^{\mathrm{2}} } \mathrm{cos}\left(\mathrm{2}{bx}\right)\:{dx}\:\:\:=\:\:\:\frac{\sqrt{\pi}}{\mathrm{2}{a}}{e}^{−{b}^{\mathrm{2}} /{a}^{\mathrm{2}} } \\ $$$$ \\ $$$$…
Question Number 122636 by mnjuly1970 last updated on 18/Nov/20 $$\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:{In}\:\:\mathrm{A}\overset{\Delta} {\mathrm{B}C}\:\:{prove}\:::\: \\ $$$$\:\:\:\ast:\:\:{sin}\left(\frac{\mathrm{A}}{\mathrm{2}}\right){sin}\left(\frac{\mathrm{B}}{\mathrm{2}}\right){sin}\left(\frac{\mathrm{C}}{\mathrm{2}}\right)\leqslant\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$……………………. \\ $$$$\:\:\:\:\ast\ast::\:\:\:{max}\left({cos}\left(\frac{\mathrm{A}}{\mathrm{2}}\right){cos}\left(\frac{\mathrm{B}}{\mathrm{2}}\right){cos}\left(\frac{\mathrm{C}}{\mathrm{2}}\right)\right)=? \\ $$$$\:\:\:\:\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:…
Question Number 57103 by turbo msup by abdo last updated on 30/Mar/19 $${let}\:{A}_{{n}} =\int\int_{{w}_{{n}} } \:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } \:\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy} \\ $$$${with}\:{w}_{{n}} =\left[\frac{\mathrm{1}}{{n}},{n}\right]×\left[\frac{\mathrm{1}}{{n}},{n}\right] \\…
Question Number 122631 by MJS_new last updated on 18/Nov/20 $$\mathrm{solve}\:\underset{\left({a}−\mathrm{1}\right)^{\mathrm{2}} } {\overset{{a}^{\mathrm{2}} } {\int}}\mathrm{cosh}^{−\mathrm{1}} \:\frac{\mathrm{1}}{\:\sqrt{{a}−\sqrt{{x}}}}\:{dx}\:\mathrm{with}\:{a}>\mathrm{0} \\ $$ Commented by liberty last updated on 18/Nov/20 $${waw}..{nice}\:{question}…
Question Number 188164 by cortano12 last updated on 26/Feb/23 Commented by cortano12 last updated on 01/Mar/23 $$\mathrm{if}\:\mathrm{x}=\mathrm{1}\Rightarrow\mathrm{f}\left(\mathrm{3}\right)+\mathrm{f}\left(\mathrm{1}\right)=\mathrm{1} \\ $$$$\mathrm{if}\:\mathrm{x}=−\mathrm{1}\Rightarrow\mathrm{f}\left(\mathrm{1}\right)+\mathrm{f}\left(\mathrm{3}\right)=−\mathrm{1} \\ $$$$\Rightarrow\mathrm{1}=−\mathrm{1}?\: \\ $$ Answered by…
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…