Question Number 208176 by mnjuly1970 last updated on 07/Jun/24 $$ \\ $$$$\:\:\:\:\:\boldsymbol{{Find}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{the}} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{{folloing}}\:\boldsymbol{{integral}}. \\ $$$$\:\:\:\:\:\:\: \\ $$$$\begin{array}{|c|}{\:\:\:\boldsymbol{\Omega}=\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \:\frac{\:\mathrm{1}}{\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\:\boldsymbol{{cosx}}}}\:\boldsymbol{{dx}}\:=\:?\:\:}\\\hline\end{array} \\ $$$$\:\:\:\:\:\:\: \\ $$ Commented…
Question Number 208178 by emilagazade last updated on 07/Jun/24 Commented by emilagazade last updated on 07/Jun/24 $${please}\:{help} \\ $$ Commented by A5T last updated on…
Question Number 208205 by efronzo1 last updated on 07/Jun/24 $$\:\:\:\int\:\left({x}^{\mathrm{3}} .\:\mathrm{5}^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}} \:\right)\:{dx}\:=? \\ $$ Answered by Frix last updated on 07/Jun/24 $$\int{x}^{\mathrm{3}} \mathrm{5}^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}}…
Question Number 208173 by Davidtim last updated on 07/Jun/24 $$\frac{\mathrm{5}}{−\infty}=? \\ $$ Commented by Davidtim last updated on 07/Jun/24 $${help}\:{me} \\ $$ Answered by Skabetix…
Question Number 208199 by efronzo1 last updated on 07/Jun/24 Answered by Frix last updated on 07/Jun/24 $${y}=\sqrt{\mathrm{18}+\mathrm{3}{x}−{x}^{\mathrm{2}} }\:\mathrm{is}\:\mathrm{a}\:\mathrm{semi}\:\mathrm{circle}\:\mathrm{with}\:{r}=\frac{\mathrm{9}}{\mathrm{2}} \\ $$$$\sqrt{{x}+\mathrm{3}}+\sqrt{\mathrm{6}−{x}}\:\mathrm{has}\:\mathrm{the}\:\mathrm{maximum}\:\begin{pmatrix}{\frac{\mathrm{3}}{\mathrm{2}}}\\{\mathrm{3}\sqrt{\mathrm{2}}}\end{pmatrix} \\ $$$$\mathrm{We}\:\mathrm{have}\:\mathrm{2}\:\mathrm{solutions}\:\mathrm{for}\:\mathrm{0}\leqslant{m}<\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{4}}\:\mathrm{and} \\ $$$$\mathrm{exactly}\:\mathrm{one}\:\mathrm{solution}\:\mathrm{at}\:{m}=\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{4}} \\…
Question Number 208194 by universe last updated on 07/Jun/24 $$\:\:\:\:\:\:\:\:{I}_{{n}} \:=\:\:\int_{\mathrm{0}\:} ^{\infty} \frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }{dx} \\ $$$$\:\:{prove}\:{that}\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{I}_{{n}} }{{n}}\:\:=\:\:\pi \\ $$ Answered by Berbere…
Question Number 208158 by necx122 last updated on 06/Jun/24 $${X},\:{Y}\:{and}\:{Z}\:{are}\:{points}\:{on}\:{the}\:{sides}\:{AB}, \\ $$$${BC}\:{and}\:{AC}\:{of}\:{the}\:{triangle}\:{ABC},\:{such} \\ $$$${that}\:{AX}:{XB}\:=\mathrm{4}:\mathrm{3},\:{BY}:{YC}=\mathrm{2}:\mathrm{3},\: \\ $$$${CZ}:{ZA}=\mathrm{2}:\mathrm{1}.\:{Find}\:{the}\:{ratio}\:{of}\:{the}\:{area} \\ $$$${of}\:{the}\:{triangle}\:{XYZ}\:{to}\:{that}\:{of}\:{triangle} \\ $$$${ABC}. \\ $$ Answered by mr…
Question Number 208148 by Blackpanther last updated on 06/Jun/24 Answered by A5T last updated on 06/Jun/24 Commented by A5T last updated on 06/Jun/24 $$\mathrm{2}{y}=\mathrm{6}{x}\Rightarrow{y}=\mathrm{3}{x} \\…
Question Number 208149 by mnjuly1970 last updated on 06/Jun/24 Answered by A5T last updated on 06/Jun/24 $$\lfloor\mathrm{2}{x}^{\mathrm{2}} \rfloor>\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}\Rightarrow{x}−\lfloor\mathrm{2}{x}^{\mathrm{2}} \rfloor<\mathrm{1}−\mathrm{2}{x}^{\mathrm{2}} +{x} \\ $$$${Suppose}\:{D}_{{f}} ,{R}_{{f}} \subseteq\mathbb{R}…
Question Number 208140 by Ghisom last updated on 06/Jun/24 $$\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{dx}}{\mathrm{1}+\sqrt[{\mathrm{3}}]{\mathrm{sin}\:{x}}}=? \\ $$$$\mathrm{exact}\:\mathrm{result}\:\mathrm{required} \\ $$ Answered by Frix last updated on 07/Jun/24 $$=\mathrm{2}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}}…