Question Number 146147 by iloveisrael last updated on 11/Jul/21 $$\:\Upsilon\:=\:\int\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{4}} \:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} }}\:=? \\ $$ Answered by EDWIN88 last updated on 11/Jul/21 $$\:{Solve}\:\int\:\frac{{dx}}{{x}^{\mathrm{4}} \:\sqrt{{x}^{\mathrm{2}} −{a}^{\mathrm{2}}…
Question Number 146131 by mathlove last updated on 11/Jul/21 Commented by iloveisrael last updated on 11/Jul/21 $$\mathrm{great}\:\mathrm{ustad} \\ $$ Answered by EDWIN88 last updated on…
Question Number 15045 by Tinkutara last updated on 07/Jun/17 $$\mathrm{In}\:\mathrm{a}\:\mathrm{curve}\:\mathrm{the}\:{x}\:\mathrm{and}\:{y}\:\mathrm{co}-\mathrm{ordinate}\:\mathrm{is} \\ $$$$\mathrm{function}\:\mathrm{of}\:{t}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}\:=\:\mathrm{cos}\:{t}\:\mathrm{and}\:{y}\:=\:\mathrm{sin}\:{t},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{for}\:{t}\:=\:\mathrm{0}\:\mathrm{to}\:{t}\:=\:\frac{\pi}{\mathrm{2}}. \\ $$ Answered by mrW1 last updated on 07/Jun/17…
Question Number 15039 by Tinkutara last updated on 07/Jun/17 $$\mathrm{The}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{an}\:\mathrm{object}\:\mathrm{is}\:\mathrm{given} \\ $$$$\mathrm{by}\:{a}\left({t}\right)\:=\:\mathrm{cos}\left(\pi{t}\right)\:\mathrm{ms}^{−\mathrm{2}} \:\mathrm{and}\:\mathrm{its}\:\mathrm{velocity} \\ $$$$\mathrm{at}\:\mathrm{time}\:{t}\:=\:\mathrm{0}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\mathrm{m}/\mathrm{s}\:\mathrm{at}\:\mathrm{origin}.\:\mathrm{Its} \\ $$$$\mathrm{velocity}\:\mathrm{at}\:{t}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{s}\:\mathrm{is}? \\ $$$$\mathrm{The}\:\mathrm{object}'\mathrm{s}\:\mathrm{position}\:\mathrm{at}\:{t}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{s}\:\mathrm{is}? \\ $$ Answered by ajfour last…
Question Number 146110 by puissant last updated on 11/Jul/21 $$\int\frac{\mathrm{x}+\mathrm{1}}{\mathrm{2x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$ Answered by gsk2684 last updated on 11/Jul/21 $$\int\frac{\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{4}{x}+\mathrm{1}\right)−\frac{\mathrm{1}}{\mathrm{4}}+\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx} \\ $$$$\frac{\mathrm{1}}{\mathrm{4}}\mathrm{log}\:\mid\mathrm{2}{x}^{\mathrm{2}} +{x}+\mathrm{1}\mid+\frac{\mathrm{3}}{\mathrm{4}}\int\frac{\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}}…
Question Number 146108 by savitar last updated on 11/Jul/21 $$\:{Solve}\:\:{in}\:\mathbb{Z}\left[{X}\right] \\ $$$$\left.\mathrm{1}\right)\:{XP}\:'\:\equiv\:−\mathrm{1}\:{mod}\left({X}^{\mathrm{4}} +\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\:{X}^{\mathrm{3}} {P}\:−{P}\:'\:\equiv\:\mathrm{1}−{X}^{\mathrm{2}} \:{mod}\left({X}^{\mathrm{4}} +\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{P}\:^{\mathrm{2}} −{X}^{\mathrm{3}} {P}−{X}^{\mathrm{2}} \:\:\equiv\:\mathrm{0}\:{mod}\left({X}^{\mathrm{2}} +\mathrm{2}\right) \\…
Question Number 15036 by Tinkutara last updated on 07/Jun/17 $$\mathrm{Rotating}\:\mathrm{a}\:\mathrm{curve}\:{y}\:=\:\sqrt{{x}}\:\mathrm{about}\:\mathrm{the}\:{x}- \\ $$$$\mathrm{axis}\:\mathrm{produces}\:\mathrm{a}\:“\mathrm{head}\:\mathrm{light}''\:\mathrm{as}\:\mathrm{shown} \\ $$$$\mathrm{below}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{disc}\:\mathrm{at}\:\mathrm{any}\:{x}? \\ $$ Commented by Tinkutara last updated on 07/Jun/17…
Question Number 15022 by Tinkutara last updated on 07/Jun/17 $$\mathrm{Evaluate}:\:\int_{\mathrm{1}} ^{\mathrm{4}} \frac{{x}^{\mathrm{2}} \:+\:{x}}{\:\sqrt{\mathrm{2}{x}\:+\:\mathrm{1}}}\:{dx}\:\left(\mathrm{Question}\:\mathrm{ID}:\right. \\ $$$$\left.\mathrm{53}\right)\:\mathrm{How}\:\mathrm{does}\:\mathrm{the}\:\mathrm{limits}\:\mathrm{change}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{solution}\:\mathrm{of}\:\mathrm{Q}.\:\mathrm{No}.\:\mathrm{53}? \\ $$ Answered by Joel577 last updated on…
Question Number 146090 by Ar Brandon last updated on 10/Jul/21 $$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sinh}\left(\mathrm{at}\right)\mathrm{sinh}\left(\mathrm{bt}\right)}{\mathrm{sinh}\left(\mathrm{ct}\right)\mathrm{e}^{\mathrm{tz}} }\mathrm{dt}= \\ $$$$\frac{\mathrm{ab}}{\mathrm{c}\left(\mathrm{z}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} +\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\mathrm{K}}}\frac{−\mathrm{4k}^{\mathrm{2}} \left(\mathrm{k}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}}…
Question Number 146073 by savitar last updated on 10/Jul/21 $${Let}\:{K}\:{be}\:{nonempty}\:\:{corps}\:,\:{K}^{\ast} ={K}−\left\{\mathrm{0}_{{K}} \right\} \\ $$$${Prove}\:{that} \\ $$$$\left.\mathrm{1}\right)\:\underset{{x}\in{K}^{\ast} } {\prod}{x}\:=\:−\mathrm{1} \\ $$$$\left.\mathrm{2}\right){Deduce}\:{that}\: \\ $$$$\:\:{p}\:{is}\:{prime}\:\Leftrightarrow\:\left({p}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[{p}\right] \\ $$ Terms…