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…
Question Number 146067 by gsk2684 last updated on 10/Jul/21 $${transform}\:{the}\:{cartesian}\:{inyegral}\: \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} {\int}}{e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} \:{dy}\:{dx}\:{into}\:{polar}\:{integral}\: \\ $$$${and}\:{evaluate}\:{it}. \\ $$ Answered…
Question Number 146062 by mnjuly1970 last updated on 10/Jul/21 $$ \\ $$$$\:\:\:\:{find}\:\:{values}\:\:{a}\:,\:{b}\:,\:{c}\:\:{such}\:{that}: \\ $$$$\:\:\:\:−\mathrm{1}\leqslant\:{ax}\:^{\mathrm{2}} +{bx}\:+{c}\:\leqslant\:\mathrm{1} \\ $$$$\:\:\:\:\:\:{and}\:\:\frac{\mathrm{6}{b}^{\:\mathrm{2}} +\:\mathrm{8}\:{a}^{\:\mathrm{2}} }{\mathrm{3}}\:{is}\:{Max}… \\ $$ Commented by mr W…
Question Number 146035 by KONE last updated on 10/Jul/21 $${help}\:{me}\:{please} \\ $$$$\int\frac{{ln}\left({x}+\mathrm{1}\right)}{{x}}{dx}=?? \\ $$$$ \\ $$ Answered by KONE last updated on 10/Jul/21 $${please} \\…
Question Number 80485 by jagoll last updated on 03/Feb/20 $${what}\:{is}\:{the}\:{king}\:\:{rule}? \\ $$ Commented by mr W last updated on 03/Feb/20 $${in}\:{mathematics}? \\ $$$${then}\:{king}\:{rule}\:{is} \\ $$$$\int_{{a}}…
Question Number 80452 by abdomathmax last updated on 03/Feb/20 $${find}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1}\right)}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$ Commented by abdomathmax last updated on 03/Mar/20 $${I}=\:{Re}\left(\int_{−\infty}…