Question Number 84505 by mathmax by abdo last updated on 13/Mar/20 $$\left.\mathrm{2}\right){calculate}\:\:\:{I}\left(\xi\right)\:=\int_{\xi} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{dx}}{\:\sqrt{\mathrm{1}+\xi{x}^{\mathrm{2}} −\sqrt{\mathrm{1}−\xi{x}^{\mathrm{2}} }}} \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{\xi\rightarrow\mathrm{0}} \:\:{I}\left(\xi\right) \\ $$ Commented by mathmax by…
Question Number 18968 by chux last updated on 02/Aug/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{side}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle} \\ $$$$\mathrm{if}\:\mathrm{side}\:\mathrm{lengths}\:\mathrm{are}\:\mathrm{consecutive}\: \\ $$$$\mathrm{integers},\mathrm{and}\:\mathrm{one}\:\mathrm{of}\:\mathrm{whose}\:\mathrm{angles} \\ $$$$\mathrm{is}\:\mathrm{twice}\:\mathrm{as}\:\mathrm{large}\:\mathrm{as}\:\mathrm{another}. \\ $$ Commented by chux last updated on 02/Aug/17…
Question Number 150037 by RoswelCod2003 last updated on 09/Aug/21 $${Random}\:{Problem}: \\ $$$$\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\left(−\mathrm{7sin}\:{x}\:+\:\mathrm{3cos}\:{x}\right)\:{dx} \\ $$$$ \\ $$$${By}\:{getting}\:{the}\:{antiderivative}\:{of}\:{the}\:{trigonometric}\:{functions}: \\ $$$$\int\:\mathrm{sin}\left({x}\right)\:{dx}\:=\:−\mathrm{cos}\:{x}\:+\:{c} \\ $$$$\int\:\mathrm{cos}\left({x}\right)\:{dx}\:=\:\mathrm{sin}\:{x}\:+\:{c} \\ $$$$=\:−\mathrm{7}\:\int\:\mathrm{sin}\:{x}\:\:+\:\:\mathrm{3}\:\int\:\mathrm{cos}\:{x}\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}}…
Question Number 18967 by Tinkutara last updated on 02/Aug/17 $$\mathrm{Let}\:\mathrm{PQRS}\:\mathrm{be}\:\mathrm{a}\:\mathrm{rectangle}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{PQ}\:=\:{a}\:\mathrm{and}\:\mathrm{QR}\:=\:{b}.\:\mathrm{Suppose}\:\mathrm{r}_{\mathrm{1}} \:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{passing}\:\mathrm{through}\:\mathrm{P} \\ $$$$\mathrm{and}\:\mathrm{Q}\:\mathrm{and}\:\mathrm{touching}\:\mathrm{RS}\:\mathrm{and}\:\mathrm{r}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{passing}\:\mathrm{through}\:\mathrm{Q} \\ $$$$\mathrm{and}\:\mathrm{R}\:\mathrm{and}\:\mathrm{touching}\:\mathrm{PS}.\:\mathrm{Show}\:\mathrm{that}\:: \\ $$$$\mathrm{5}\left({a}\:+\:{b}\right)\:\leqslant\:\mathrm{8}\left(\mathrm{r}_{\mathrm{1}} \:+\:\mathrm{r}_{\mathrm{2}}…
Question Number 150039 by bobhans last updated on 09/Aug/21 $$\mathrm{Prove}\:\mathrm{that}\:\frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{c}}+\frac{\mathrm{c}}{\mathrm{a}}\geqslant\sqrt{\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{1}}{\mathrm{b}^{\mathrm{2}} +\mathrm{1}}}+\sqrt{\frac{\mathrm{b}^{\mathrm{2}} +\mathrm{1}}{\mathrm{c}^{\mathrm{2}} +\mathrm{1}}}+\sqrt{\frac{\mathrm{c}^{\mathrm{2}} +\mathrm{1}}{\mathrm{a}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}\: \\ $$ Answered by EDWIN88 last updated…
Question Number 18965 by Tinkutara last updated on 02/Aug/17 $$\mathrm{Two}\:\mathrm{blocks}\:\mathrm{of}\:\mathrm{masses}\:{M}\:\mathrm{and}\:\mathrm{2}{M}\:\mathrm{are} \\ $$$$\mathrm{connected}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}\:\mathrm{through}\:\mathrm{a}\:\mathrm{light} \\ $$$$\mathrm{spring}\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{figure}.\:\mathrm{If}\:\mathrm{we}\:\mathrm{push}\:\mathrm{the} \\ $$$$\mathrm{mass}\:{M}\:\mathrm{with}\:\mathrm{a}\:\mathrm{force}\:{F}\:\mathrm{which}\:\mathrm{cause} \\ $$$$\mathrm{acceleration}\:\mathrm{a}\:\mathrm{in}\:\mathrm{mass}\:{M},\:\mathrm{what}\:\mathrm{will}\:\mathrm{the} \\ $$$$\mathrm{acceleration}\:\mathrm{in}\:\mathrm{2}{M}? \\ $$ Commented by Tinkutara…
Question Number 150033 by cherokeesay last updated on 08/Aug/21 Answered by maged last updated on 09/Aug/21 $${A}_{{c}} =\frac{\pi{R}^{\mathrm{2}} }{\mathrm{2}}=\frac{\mathrm{36}\pi}{\mathrm{2}}=\mathrm{18}\pi \\ $$$$\mid{BH}\mid={R}\mathrm{sin}\:\mathrm{60}°=\mathrm{6}×\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}=\mathrm{3}\sqrt{\mathrm{3}} \\ $$$${A}_{{y}} ={A}_{{s}} +{A}_{{t}}…
Question Number 84498 by M±th+et£s last updated on 13/Mar/20 $$\int\sqrt{{x}}\:{cos}\sqrt{{x}}\:{dx} \\ $$ Commented by jagoll last updated on 13/Mar/20 $$\int\:\frac{{x}\:\mathrm{cos}\:\sqrt{{x}}}{\:\sqrt{{x}}}\:{dx}\: \\ $$$${let}\:\sqrt{{x}}\:=\:{t}\:\Rightarrow\:{x}={t}^{\mathrm{2}} \\ $$$${dx}\:=\:\mathrm{2}{t}\:{dt} \\…
Question Number 18963 by Tinkutara last updated on 02/Aug/17 $$\mathrm{Two}\:\mathrm{blocks}\:{A}\:\mathrm{and}\:{B}\:\mathrm{each}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{1}\:\mathrm{kg} \\ $$$$\mathrm{are}\:\mathrm{placed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{smooth}\:\mathrm{horizontal} \\ $$$$\mathrm{surface}.\:\mathrm{Two}\:\mathrm{horizontal}\:\mathrm{forces}\:\mathrm{5}\:\mathrm{N}\:\mathrm{and} \\ $$$$\mathrm{10}\:\mathrm{N}\:\mathrm{are}\:\mathrm{applied}\:\mathrm{on}\:\mathrm{the}\:\mathrm{blocks}\:{A}\:\mathrm{and}\:{B} \\ $$$$\mathrm{respectively}\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{figure}.\:\mathrm{The} \\ $$$$\mathrm{block}\:{A}\:\mathrm{does}\:\mathrm{not}\:\mathrm{slide}\:\mathrm{on}\:\mathrm{block}\:{B}.\:\mathrm{Then} \\ $$$$\mathrm{the}\:\mathrm{normal}\:\mathrm{reaction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{block}\:\mathrm{is} \\…
Question Number 18962 by Tinkutara last updated on 02/Aug/17 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{sin}^{\mathrm{5}} \:\theta\:+\:\frac{\mathrm{1}}{\mathrm{sin}\:\theta}\:=\:\frac{\mathrm{1}}{\mathrm{cos}\:\theta}\:+\:\mathrm{cos}^{\mathrm{5}} \:\theta\:\mathrm{where} \\ $$$$\theta\:\in\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\right)\:,\:\mathrm{is} \\ $$ Answered by behi.8.3.4.1.7@gmail.com last updated on 02/Aug/17…