Question Number 146746 by mathdanisur last updated on 15/Jul/21 $$\underset{\:\mathrm{0}} {\overset{\:\mathrm{4}} {\int}}\:\sqrt{\mathrm{16}\:-\:{x}^{\mathrm{2}} }\:{dx}\:=\:? \\ $$ Answered by qaz last updated on 15/Jul/21 $$\int_{\mathrm{0}} ^{\mathrm{4}} \sqrt{\mathrm{16}−\mathrm{x}^{\mathrm{2}}…
Question Number 146744 by mathdanisur last updated on 15/Jul/21 $${if}\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}{f}\left({x}\right){dx}\:=\:\mathrm{7}\:\:\:{and}\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}\mathrm{4}\:{g}\left({x}\right){dx}\:=\:−\mathrm{6} \\ $$$${find}\:\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}\left(\mathrm{3}\:{f}\left({x}\right)−\mathrm{8}\:{g}\left({x}\right)\right)\:{dx}\:=\:? \\ $$ Answered by liberty last updated…
Question Number 81206 by Power last updated on 10/Feb/20 Commented by mathmax by abdo last updated on 10/Feb/20 $${let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)}\:{x}^{{n}} \:{with}\:\mid{x}\mid\leqslant\mathrm{1}\:\:{and}\:{x}\neq−\mathrm{1} \\ $$$${f}\left({x}\right)=\sum_{{n}=\mathrm{0}}…
Question Number 146730 by puissant last updated on 15/Jul/21 Commented by puissant last updated on 15/Jul/21 $$\mathrm{merci} \\ $$ Commented by Olaf_Thorendsen last updated on…
Question Number 15656 by Tinkutara last updated on 12/Jun/17 $$\mathrm{Solve}\::\:\mathrm{0}\:\leqslant\:{x}^{\mathrm{2}} \:−\:\mathrm{5}{x}\:+\:\mathrm{7}\:<\:\mathrm{1} \\ $$ Answered by ajfour last updated on 12/Jun/17 $$\:\:\:\:\:\mathrm{0}\leqslant\:\left({x}−\frac{\mathrm{5}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{7}−\frac{\mathrm{25}}{\mathrm{4}}\:<\:\mathrm{1} \\ $$$$\:\:\:\:\:\mathrm{0}\leqslant\:\left({x}−\frac{\mathrm{5}}{\mathrm{2}}\right)^{\mathrm{2}} +\frac{\mathrm{3}}{\mathrm{4}}\:<\:\mathrm{1}…
Question Number 146725 by mathdanisur last updated on 15/Jul/21 $${cos}\left({x}\right)\:\centerdot\:{cos}\left(\mathrm{3}{x}\right)\:=\:{cos}\left(\mathrm{5}{x}\right)\:\centerdot\:{cos}\left(\mathrm{7}{x}\right) \\ $$$$\Rightarrow\:{x}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 15/Jul/21 $$\mathrm{cos}{x}.\mathrm{cos3}{x}\:=\:\mathrm{cos5}{x}.\mathrm{cos7}{x} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{cos}\left({x}−\mathrm{3}{x}\right)+\mathrm{cos}\left({x}+\mathrm{3}{x}\right)\right]\:= \\…
Question Number 146680 by mathdanisur last updated on 14/Jul/21 $${Compare}:\:\:\mathrm{100}^{\mathrm{101}} \:\:{and}\:\:\:\mathrm{101}^{\mathrm{100}} \\ $$ Answered by Olaf_Thorendsen last updated on 14/Jul/21 $$\mathrm{log}_{\mathrm{10}} \mathrm{101}\:=\:\mathrm{2}+\mathrm{log}_{\mathrm{10}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{100}}\right) \\ $$$$\mathrm{100log}_{\mathrm{10}}…
Question Number 146672 by mathdanisur last updated on 14/Jul/21 Answered by Olaf_Thorendsen last updated on 15/Jul/21 $$\mathrm{S}\:=\:\frac{\underset{{n}=\mathrm{1}} {\overset{{x}} {\sum}}{n}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }{\underset{{n}=\mathrm{1}} {\overset{{x}} {\sum}}{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)} \\ $$$$\mathrm{S}\:=\:\frac{\underset{{n}=\mathrm{1}}…
Question Number 146678 by mathdanisur last updated on 14/Jul/21 $$\int\:\sqrt{\mathrm{2}\:+\:{x}^{\mathrm{2}} }\:{dx}\:=\:? \\ $$ Answered by Ar Brandon last updated on 14/Jul/21 $$\mathrm{I}=\int\sqrt{\mathrm{2}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$$$\:\:=\mathrm{2}\int\sqrt{\mathrm{1}+\mathrm{sinh}^{\mathrm{2}}…
Question Number 146677 by mathdanisur last updated on 14/Jul/21 $${if}\:\:\:{f}\left({x}\right)\:=\:\mathrm{3}^{\boldsymbol{{x}}+\mathrm{1}} \:\:\:{find}\:\:\:\frac{{f}\left(\mathrm{2}{x}\:+\:\mathrm{1}\right)}{{f}\left({x}\:+\:\mathrm{1}\right)}\:=\:? \\ $$ Answered by hknkrc46 last updated on 14/Jul/21 $$\blacktriangleright\:\boldsymbol{{f}}\left(\mathrm{2}\boldsymbol{{x}}\:+\:\mathrm{1}\right)\:=\:\mathrm{3}^{\left(\mathrm{2}\boldsymbol{{x}}\:+\:\mathrm{1}\right)\:+\:\mathrm{1}} \:=\:\mathrm{3}^{\mathrm{2}\boldsymbol{{x}}\:+\:\mathrm{2}} \\ $$$$\blacktriangleright\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\:+\:\mathrm{1}\right)\:=\:\mathrm{3}^{\left(\boldsymbol{{x}}\:+\:\mathrm{1}\right)\:+\:\mathrm{1}} \:=\:\mathrm{3}^{\boldsymbol{{x}}\:+\:\mathrm{2}}…