Question Number 61801 by aliesam last updated on 08/Jun/19 $$\underset{\mathrm{2}\pi} {\overset{\mathrm{4}\pi} {\int}}\sqrt{\mathrm{1}−{cos}\left({x}\right)}\:{dx} \\ $$ Commented by MJS last updated on 08/Jun/19 $$\sqrt{\mathrm{1}−\mathrm{cos}\:{x}}=\sqrt{\mathrm{2}}\mathrm{sin}\:\frac{{x}}{\mathrm{2}} \\ $$$$\mathrm{now}\:\mathrm{it}'\mathrm{s}\:\mathrm{easy} \\…
Question Number 127337 by ajfour last updated on 28/Dec/20 Commented by ajfour last updated on 28/Dec/20 Commented by ajfour last updated on 28/Dec/20 $${Any}\:{construction}\:{method}\:{forthis}? \\…
Question Number 61799 by alphaprime last updated on 08/Jun/19 $$\mathrm{If}\:\mathrm{a}\:=\:\mathrm{Cos}\alpha\:−\mathrm{iSin}\alpha\:\mathrm{and}\:\mathrm{b}\:=\:\mathrm{Cos}\beta\:−\mathrm{iSin}\beta \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\frac{\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{1}−\mathrm{ab}\right)}{\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{1}+\mathrm{ab}\right)}\:=\:\frac{\mathrm{Sin}\alpha+\mathrm{Sin}\beta}{\mathrm{Sin}\alpha−\mathrm{Sin}\beta} \\ $$ Answered by tanmay last updated on 09/Jun/19 $${a}={e}^{−{i}\alpha} \:\:\:{b}={e}^{−{i}\beta} \\ $$$$\frac{{a}}{{b}}={e}^{−{i}\left(\propto−\beta\right)}…
Question Number 127332 by snipers237 last updated on 28/Dec/20 $${let}\:{a},{b},{c}\:{be}\:{integers}\:{such}\:{as} \\ $$$$\mathrm{0}<{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{abc}\:<\:{c}\: \\ $$$${prove}\:{that}\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{abc}}\:\:{is}\:{an}\:{integer} \\ $$ Terms of Service Privacy Policy…
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Question Number 192867 by beto last updated on 30/May/23 $${derivate}\:{of}\:\:{csc}\left(\mathrm{2}{x}\right)\:{by}\:\:{definition} \\ $$ Answered by cortano12 last updated on 02/Jun/23 $$\:\frac{\mathrm{d}\left(\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{2x}}\right)}{\mathrm{dx}}=\underset{\mathrm{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{sin}\:\left(\mathrm{2x}+\mathrm{2h}\right)}−\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{2x}}}{\mathrm{h}} \\ $$$$\:=\:\underset{\mathrm{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{2x}−\mathrm{sin}\:\left(\mathrm{2x}+\mathrm{2h}\right)}{\mathrm{h}\:\mathrm{sin}\:\mathrm{2x}\:\mathrm{sin}\:\left(\mathrm{2x}+\mathrm{2h}\right)} \\…
Question Number 61791 by naka3546 last updated on 08/Jun/19 $$\underset{{x}\:\rightarrow\:\infty} {\mathrm{lim}}\:\:\:\frac{\mathrm{6}^{{x}} }{\mathrm{2}^{{x}} \:+\:\mathrm{4}^{{x}} }\:\:\:=\:\:\:\infty\:\:\:\:\:? \\ $$ Commented by mr W last updated on 08/Jun/19 $$\underset{{x}\:\rightarrow\:\infty}…
Question Number 192862 by Red1ight last updated on 30/May/23 $$\mathrm{Solve}\:\mathrm{for}\:{x} \\ $$$${x}^{\mathrm{3}} −\mathrm{115}{x}+\mathrm{150}=\mathrm{0} \\ $$ Answered by Frix last updated on 30/May/23 $$\left({x}−\mathrm{10}\right)\left({x}^{\mathrm{2}} +\mathrm{10}{x}−\mathrm{15}\right)=\mathrm{0} \\…
Question Number 127322 by MathSh last updated on 28/Dec/20 $${f}\left({x}\right)\:{function}\:{pair}\:{and}\:\left(−\infty;\mathrm{0}\right] \\ $$$${decreases}\:{in}\:{the}\:{range} \\ $$$${f}\left({x}\right)\geqslant{f}\left(−\mathrm{3}\right)\:{solve}\:{the}\:{inequality} \\ $$ Answered by mindispower last updated on 28/Dec/20 $$\left.\right]\left.−\infty,−\mathrm{3}\right]\cup\left[\mathrm{3},+\infty\left[\right.\right. \\…
Question Number 192856 by universe last updated on 29/May/23 $$\:{if}\:{a}+{b}+{c}\:+{d}\:\:=\:\mathrm{63}\:{and}\:{a},{b},{c},{d}\:\in\:\mathbb{N}\:{find}\: \\ $$$$\:\:\:{the}\:{maximum}\:{value}\:{of}\:{ab}+{bc}+{cd}\:=\:? \\ $$ Answered by Frix last updated on 29/May/23 $$\mathrm{32}+\mathrm{31}+\mathrm{0}+\mathrm{0}=\mathrm{63} \\ $$$$\mathrm{32}×\mathrm{31}=\mathrm{992} \\…