Question Number 137328 by mnjuly1970 last updated on 01/Apr/21 Answered by Dwaipayan Shikari last updated on 01/Apr/21 $$−\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {tan}^{\mathrm{2}} \left({x}\right){log}\left({sinx}\right){dx}\:\:\:\:\:\: \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}\left({sinx}\right)−\int_{\mathrm{0}}…
Question Number 6257 by 314159 last updated on 20/Jun/16 $${Given}\:{that}\:{a}\:{and}\:{b}\:{are}\:{positive}\:{real}\:{number} \\ $$$${such}\:{that}\:{b}<\mathrm{4}{a}+\mathrm{1},{show}\:{that}\:\frac{\mathrm{2}{a}+{b}}{\mathrm{4}{a}+\mathrm{1}}<\sqrt{\mathrm{4}{a}^{\mathrm{2}} +{b}}\:. \\ $$ Answered by Yozzii last updated on 20/Jun/16 $$\frac{\mathrm{2}{a}+{b}}{\mathrm{4}{a}+\mathrm{1}}<\sqrt{\mathrm{4}{a}^{\mathrm{2}} +{b}}\:\:\:\:\:\:{a},{b}>\mathrm{0} \\…
Question Number 71790 by jatin123 last updated on 20/Oct/19 Answered by $@ty@m123 last updated on 20/Oct/19 $$=\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{3}}{\mathrm{4}}×…..×\frac{\mathrm{98}}{\mathrm{99}}×\frac{\mathrm{99}}{\mathrm{100}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{100}}=\frac{\mathrm{1}}{\mathrm{50}} \\ $$ Commented by jatin123 last…
Question Number 137324 by greg_ed last updated on 01/Apr/21 $$\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{evaluate}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{one}}\:: \\ $$$$\mathrm{P}\:=\:\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{1958}}\right)\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{1959}}\right)\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{1960}}\right)…\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{2017}}\right)\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{2018}}\right)\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{2019}}\right) \\ $$$$\boldsymbol{\mathrm{P}}\:=\:?\: \\ $$ Answered by som(math1967) last updated on 01/Apr/21 $${P}=\left(\frac{\mathrm{1959}}{\mathrm{1958}}\right)\left(\frac{\mathrm{1960}}{\mathrm{1959}}\right)\left(\frac{\mathrm{1961}}{\mathrm{1960}}\right)..\left(\frac{\mathrm{2019}}{\mathrm{2018}}\right)\left(\frac{\mathrm{2020}}{\mathrm{2019}}\right) \\…
Question Number 137327 by mr W last updated on 01/Apr/21 $${in}\:{triangle}\:\Delta{ABC}:\:{BC}=\mathrm{1},\:\angle{B}=\mathrm{2}\angle{A}. \\ $$$${find}\:{the}\:{maximum}\:{area}\:{of}\:\Delta{ABC}. \\ $$ Answered by EDWIN88 last updated on 01/Apr/21 $$\angle\mathrm{A}+\angle\mathrm{B}+\angle\mathrm{C}\:=\pi\:;\:\mathrm{3}\angle\mathrm{A}+\angle\mathrm{C}=\pi \\ $$$$\mathrm{let}\:\angle\mathrm{A}\:=\alpha\:;\:\angle\mathrm{B}=\mathrm{2}\alpha\:;\:\angle\mathrm{C}=\pi−\mathrm{3}\alpha…
Question Number 137321 by mathocean1 last updated on 01/Apr/21 $${H}\left({x}\right)=\frac{\mathrm{8}}{{x}−\mathrm{2}}\:{and}\:{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{6}}{\mathrm{2}{x}−\mathrm{4}} \\ $$$$\left.\mathrm{1}\right){Calculate}\:{the}\:{surface}\:{V}_{\boldsymbol{{n}}} \:{of} \\ $$$${area}\:{limited}\:{by}\:{the}\:{the}\:{line} \\ $$$${x}=\mathrm{6};\:{x}=\mathrm{6}+\boldsymbol{{n}}\:\left({n}\in\mathbb{N}^{\ast} \right)\:{and}\:{the} \\ $$$${curve}\:{of}\:{H}\left({x}\right)\:{and}\:{f}\left({x}\right)\:{in}\:{function} \\ $$$${of}\:\boldsymbol{{n}}. \\ $$$$\left.\mathrm{2}\right)\:{Knowing}\:{that}\:\mathrm{1}^{\mathrm{2}}…
Question Number 6251 by net last updated on 20/Jun/16 $$\mathrm{3}{x}−\mathrm{2}{y}−\mathrm{4}{z}=\mathrm{2} \\ $$$$\mathrm{3}{y}−\mathrm{4}{z}=−\mathrm{2} \\ $$$$\mathrm{2}{y}+\mathrm{6}{z}=−\mathrm{1} \\ $$ Answered by Rasheed Soomro last updated on 20/Jun/16 $$\mathrm{3}{x}−\mathrm{2}{y}−\mathrm{4}{z}=\mathrm{2}………\left({i}\right)…
Question Number 6248 by FilupSmith last updated on 20/Jun/16 $$\int_{−\infty} ^{\:\infty} {e}^{−{u}} {u}^{{n}} {du}\:=\:??? \\ $$ Commented by 123456 last updated on 20/Jun/16 $$\mathrm{i}\:\mathrm{think}\:\mathrm{that}\:\mathrm{it}\:\mathrm{diverges} \\…
Question Number 137317 by physicstutes last updated on 01/Apr/21 Commented by physicstutes last updated on 01/Apr/21 $$\mathrm{The}\:\mathrm{figure}\:\mathrm{above}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{composite}\:\mathrm{bar} \\ $$$$\mathrm{made}\:\mathrm{of}\:\mathrm{two}\:\mathrm{materials}\:{X}\:\mathrm{and}\:{Y}.\:\mathrm{The}\:\mathrm{end} \\ $$$$\mathrm{of}\:{X}\:\mathrm{is}\:\mathrm{maintained}\:\mathrm{at}\:\mathrm{100}°\mathrm{C}\:\mathrm{and}\:\mathrm{it}\:\mathrm{has} \\ $$$$\mathrm{length}\:\mathrm{of}\:\mathrm{25}\:\mathrm{cm}\:\mathrm{while}\:\mathrm{that}\:\mathrm{of}\:{Y}\:\mathrm{is}\:\mathrm{maintained} \\ $$$$\mathrm{at}\:\mathrm{0}°\mathrm{C}\:\mathrm{with}\:\mathrm{lenght}\:\mathrm{75}\:\mathrm{cm}.\:\mathrm{Find}\:\mathrm{the}…
Question Number 6247 by Yozzii last updated on 20/Jun/16 $${Show}\:{that},\:\forall{x}\in\mathbb{R},\:{the}\:{sequence}\:\left\{{f}\left({n}\right)\right\}_{\mathrm{0}} ^{\infty} \\ $$$${defined}\:{by}\:{f}\left(\mathrm{0}\right)={cosx},\:{f}\left(\mathrm{1}\right)={sin}\left({cosx}\right)\:{and} \\ $$$${f}\left({n}\right)=\begin{cases}{{sin}\left({cos}\left({f}\left({n}−\mathrm{2}\right)\right)\right)\:\:{if}\:{n}\geqslant\mathrm{3}\:{is}\:{odd}}\\{{cos}\left({sin}\left({f}\left({n}−\mathrm{2}\right)\right)\right)\:\:{if}\:{n}\geqslant\mathrm{2}\:{is}\:{even},}\end{cases} \\ $$$${converges}\:{to}\:{a}\:{limit}\:{l}\in\left(\mathrm{0}.\mathrm{6},\mathrm{0}.\mathrm{7}\right). \\ $$$${If}\:{you}\:{can},\:{determine}\:{the}\:{exact}\:{value} \\ $$$${of}\:{l}. \\ $$ Commented by…