Question Number 4156 by Yozzii last updated on 30/Dec/15 $${Let}\:{I}\left({n}\right)=\int_{\mathrm{0}} ^{{n}} \left\{\sqrt{{a}+{coshx}}−\sqrt{{a}+{sinhx}}\right\}{dx} \\ $$$${with}\:\mathrm{0}<{a}<\mathrm{1}\:{and}\:{define}\:{I}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}{I}\left({n}\right). \\ $$$${Does}\:{I}\:{exist}?\: \\ $$ Commented by 123456 last updated on…
Question Number 4140 by prakash jain last updated on 29/Dec/15 $$\mathrm{Four}\:\mathrm{persons}\:\mathrm{a},\:\mathrm{b},\:\mathrm{c},\:\mathrm{d}\:\mathrm{are}\:\mathrm{standing}\:\mathrm{at}\:\mathrm{four} \\ $$$$\mathrm{vertices}\:\mathrm{of}\:\mathrm{square}\:\mathrm{ABCD}. \\ $$$$\mathrm{All}\:\mathrm{four}\:\mathrm{start}\:\mathrm{moving}\:\mathrm{simultaneously}\:\mathrm{such} \\ $$$$\boldsymbol{\mathrm{a}}\:\mathrm{is}\:\mathrm{always}\:\mathrm{moving}\:\mathrm{towards}\:\boldsymbol{\mathrm{b}}\:\mathrm{on}\:\mathrm{a}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{between}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\boldsymbol{\mathrm{b}}.\:\mathrm{Similary}\:\boldsymbol{\mathrm{b}}\:\mathrm{is}\:\mathrm{always} \\ $$$$\mathrm{moving}\:\mathrm{directly}\:\mathrm{towards}\:\boldsymbol{\mathrm{c}},\:\boldsymbol{\mathrm{c}}\:\mathrm{is}\:\mathrm{directly} \\ $$$$\mathrm{moving}\:\mathrm{towards}\:\boldsymbol{\mathrm{d}}\:\mathrm{and}\:\boldsymbol{\mathrm{d}}\:\mathrm{is}\:\mathrm{directly}\:\mathrm{moving} \\ $$$$\mathrm{towards}\:\boldsymbol{\mathrm{a}}.…
Question Number 135215 by mnjuly1970 last updated on 11/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..{calculus}\:{preliminary}…. \\ $$$$\:\:\:{Q}:\:{f}\left({x}\right)=\mathrm{2}^{{x}} −\mathrm{2}^{−{x}} \:\Rightarrow\:{f}^{\:−\mathrm{1}} \left({x}\right)=??? \\ $$$$\:\:{solution}: \\ $$$$\:\:\:\:\:{y}=\mathrm{2}^{{x}} −\mathrm{2}^{−{x}} \:\:\:….. \\ $$$$\:\:\:\:\:\:{y}=\frac{\mathrm{2}^{\mathrm{2}{x}} −\mathrm{1}}{\mathrm{2}^{{x}} }\:\Rightarrow\mathrm{2}^{\mathrm{2}{x}}…
Question Number 4133 by prakash jain last updated on 29/Dec/15 $$\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{neccesary}\:\mathrm{and}\:\mathrm{sufficient} \\ $$$$\mathrm{conditions}\:\mathrm{so}\:\mathrm{that} \\ $$$$\int_{−\infty} ^{\:+\infty} \left[\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{f}\left({n},{x}\right)\right]{dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\int_{−\infty} ^{+\infty} {f}\left({n},{x}\right){dx}\right] \\ $$…
Question Number 4116 by Filup last updated on 29/Dec/15 $$\mathrm{For}:\:{f}\left({x}\right)=\mid{ax}^{{n}} +{b}\mid \\ $$$$\mathrm{when}\:{f}\left(\alpha\right)\:\mathrm{o}\:{f}\left(\beta\right)\:\mathrm{is}\:\mathrm{continuous}, \\ $$$$\mathrm{Does}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{a}\:\mathrm{solution}: \\ $$$${S}=\int_{\alpha} ^{\:\beta} {f}\left({x}\right){dx} \\ $$$$\alpha<\beta \\ $$ Commented by…
Question Number 135174 by bemath last updated on 11/Mar/21 $$\mathcal{Z}\:=\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{arctan}\:\left(\mathrm{sin}\:\mathrm{x}\right)\:\mathrm{dx}\:+\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{4}} \mathrm{arcsin}\:\left(\mathrm{tan}\:\mathrm{x}\right)\:\mathrm{dx} \\ $$ Answered by john_santu last updated on 11/Mar/21 $${let}\:\mathcal{Z}_{\mathrm{1}} =\int_{\mathrm{0}}…
Question Number 69637 by MJS last updated on 26/Sep/19 $$…\mathrm{now}\:\mathrm{try}\:\mathrm{this}\:\mathrm{one}: \\ $$$$\int\frac{{dx}}{{x}^{\mathrm{1}/\mathrm{2}} −{x}^{\mathrm{1}/\mathrm{3}} −{x}^{\mathrm{1}/\mathrm{6}} }= \\ $$ Answered by Kunal12588 last updated on 26/Sep/19 $${t}={x}^{\mathrm{1}/\mathrm{6}}…
Question Number 69623 by aliesam last updated on 25/Sep/19 $$\int\frac{\mathrm{1}}{\:\sqrt{{x}}\:+\:\sqrt[{\mathrm{3}}]{{x}}}\:{dx} \\ $$ Answered by MJS last updated on 25/Sep/19 $$\mathrm{we}\:\mathrm{had}\:\mathrm{this}\:\mathrm{before}… \\ $$$$\int\frac{{dx}}{{x}^{\mathrm{1}/\mathrm{2}} +{x}^{\mathrm{1}/\mathrm{3}} }= \\…
Question Number 4071 by Yozzii last updated on 27/Dec/15 Commented by Yozzii last updated on 27/Dec/15 $${Just}\:{sharing}\:{some}\:{problems}\:{I}\:{did} \\ $$$${today}. \\ $$ Terms of Service Privacy…
Question Number 69597 by ozodbek last updated on 25/Sep/19 Commented by mathmax by abdo last updated on 25/Sep/19 $${generally}\:{let}\:{find}\:{f}\left({a}\right)\:=\int\sqrt{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$${changement}\:{x}\:={ash}\left({t}\right)\:{give}\:{f}\left({a}\right)=\int{ach}\left({t}\right){ach}\left({t}\right){dt} \\ $$$$={a}^{\mathrm{2}}…