Question Number 146865 by mathdanisur last updated on 16/Jul/21 $${arcsin}\left({x}^{\mathrm{2}} −\mathrm{4}\right)\:=\:{arcsin}\left(\mathrm{2}{x}\:+\:\mathrm{4}\right) \\ $$$$\Rightarrow\:{x}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 16/Jul/21 $$\mathrm{arcsin}\left({x}^{\mathrm{2}} −\mathrm{4}\right)\:=\:\mathrm{arcsin}\left(\mathrm{2}{x}+\mathrm{4}\right) \\…
Question Number 146860 by mathdanisur last updated on 16/Jul/21 $$\int{sin}\left({x}\right)\:{cos}\left({x}\right)\:{dx}\:=\:? \\ $$ Answered by puissant last updated on 16/Jul/21 $$=\int\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\left(\mathrm{2x}\right)\mathrm{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\left(\mathrm{2x}\right)\right]+\mathrm{k} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{cos}\left(\mathrm{2x}\right)+\mathrm{k}. \\…
Question Number 146866 by mathdanisur last updated on 16/Jul/21 $$\frac{{sin}^{\mathrm{6}} \boldsymbol{\alpha}\:+\:{cos}^{\mathrm{6}} \boldsymbol{\alpha}\:-\:\mathrm{1}}{{sin}^{\mathrm{4}} \boldsymbol{\alpha}\:-\:{sin}^{\mathrm{2}} \boldsymbol{\alpha}}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 16/Jul/21 $${a}\:=\:\mathrm{sin}^{\mathrm{6}} \alpha+\mathrm{cos}^{\mathrm{6}}…
Question Number 146859 by mathdanisur last updated on 16/Jul/21 $$\underset{\:\mathrm{0}} {\overset{\mathrm{3}} {\int}}\sqrt{\left({x}+\mathrm{2}\right)^{\mathrm{2}} −\mathrm{8}{x}}\:=\:? \\ $$ Answered by mathmax by abdo last updated on 16/Jul/21 $$\mathrm{I}=\int_{\mathrm{0}}…
Question Number 146854 by mathdanisur last updated on 16/Jul/21 $${Solve}\:{for}\:{real}\:{numbers}\:{the}\:{following} \\ $$$${system}\:{of}\:{equations} \\ $$$$\begin{cases}{{a}\left({a}+\mathrm{1}\right)\:=\:{b}−\mathrm{1}}\\{{a}^{\mathrm{2}} \left({b}+\mathrm{3}\right)+\mathrm{2}{a}\:=\:−\mathrm{1}}\end{cases} \\ $$ Commented by Mrsof last updated on 16/Jul/21 $${a}\left({a}+{b}\right)+\mathrm{4}={b}+\mathrm{3}…
Question Number 81301 by mr W last updated on 11/Feb/20 Commented by mr W last updated on 11/Feb/20 $${Find}\:{f}\left({x}\right)=? \\ $$$$\left({x}\in\mathbb{N}\right) \\ $$ Commented by…
Question Number 146817 by mathdanisur last updated on 15/Jul/21 Answered by mindispower last updated on 15/Jul/21 $$\Leftrightarrow\frac{{y}\left({x}+{y}\right)+{z}\left({x}+{z}\right)}{\:\sqrt{\left({x}+{y}\right)\left({x}+{z}\right)}}\geqslant{y}+{z}…. \\ $$$${y}\left({x}+{y}\right)+{z}\left({x}+{z}\right)={x}\left({y}+{z}\right)+{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \geqslant{x}\left({y}+{z}\right)+\frac{\mathrm{1}}{\mathrm{2}}\left({y}+{z}\right)^{\mathrm{2}} \\ $$$$=\left({y}+{z}\right)\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\left({y}+{z}\right)\right)=\left({y}+{z}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}\left(\left({x}+{y}\right)+\left({x}+{z}\right)\right)\right) \\ $$$$\geqslant\left({y}+{z}\right).\frac{\mathrm{1}}{\mathrm{2}}.\mathrm{2}\sqrt{\left({x}+{y}\right)\left({x}+{z}\right)}=\left({y}+{z}\right)\sqrt{\left({x}+{y}\right)\left({x}+{z}\right)}…
Question Number 15721 by ajfour last updated on 13/Jun/17 $${If}\:\:\:\:\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}=\mathrm{1} \\ $$$$\:\:\:\:\:{with}\:\mathrm{0}<{x},\:{y},\:{z}\:<\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:{then}\:{find}\:{tbe}\:\:{range}\:{of}\:{values}\:{of} \\ $$$$\:\:\:\frac{\mathrm{1}}{\boldsymbol{{x}}+\boldsymbol{{y}}}+\frac{\mathrm{1}}{\boldsymbol{{y}}+\boldsymbol{{z}}}+\frac{\mathrm{1}}{\boldsymbol{{z}}+\boldsymbol{{x}}}\:. \\ $$ Answered by mrW1 last updated on 13/Jun/17…
Question Number 81251 by mr W last updated on 10/Feb/20 Commented by mind is power last updated on 10/Feb/20 $${xyln}\left({x}\right)={ln}\left({y}\right) \\ $$$${xyln}\left({y}\right)=\mathrm{4}{ln}\left({x}\right) \\ $$$$\Leftrightarrow \\…
Question Number 146782 by mathdanisur last updated on 15/Jul/21 $${Find}\:{the}\:{modulus}\:{of}\:{a}\:{complex} \\ $$$${number}: \\ $$$${Z}\:=\:{cos}\:\mathrm{40}\:+\:{i}\:{sin}\:\mathrm{40}\:+\mathrm{1}\:=\:? \\ $$ Answered by Ar Brandon last updated on 15/Jul/21 $$\mathrm{z}=\mathrm{e}^{\mathrm{40i}}…