Question Number 6136 by sanusihammed last updated on 15/Jun/16 $${Show}\:{that}\:{of}\:{all}\:{rectangles}\:{inscribed}\:{in}\:{a}\:{given}\:{circle}\: \\ $$$${the}\:{square}\:{has}\:{a}\:{maximum}\:{area}. \\ $$ Answered by Rasheed Soomro last updated on 15/Jun/16 $${All}\:{the}\:{rectangles}\:{inscribed}\:{in}\:{same}\:{circle} \\ $$$${have}\:{equal}\:{diagonals}\:{and}\:{vice}\:{versa}.…
Question Number 6135 by sanusihammed last updated on 15/Jun/16 $${Of}\:{all}\:{rectangular}\:{boxes}\:{without}\:{a}\:{lid}\:{and}\:{having}\:{a}\:{given}\: \\ $$$${surface}\:{area}\:.\:{Find}\:{the}\:{one}\:{with}\:{maximum}\:{volume}. \\ $$ Commented by FilupSmith last updated on 15/Jun/16 $$\mathrm{Edge}\:\mathrm{lengths}\:{a},\:{b},\:{c} \\ $$$$\mathrm{Max}\:\mathrm{volume}\:\mathrm{when}\:{a}={b}={c} \\…
Question Number 6133 by sanusihammed last updated on 15/Jun/16 $${Find}\:{all}\:{positive}\:{integers}\:{n}\:{for}\:{which}\:{there}\:{exist}\:{non}\:{negative} \\ $$$${integer}\:\:{a}_{\mathrm{1}\:} ,\:{a}_{\mathrm{2}} \:,\:{a}_{\mathrm{3}} \:,\:\:…..\:,\:{a}_{{n}\:} .\:\:\:{Such}\:{that}\:. \\ $$$$\frac{\mathrm{1}}{\mathrm{2}^{{a}_{\mathrm{1}} } }+\frac{\mathrm{1}}{\mathrm{2}^{{a}_{\mathrm{2}} } }+…..+\frac{\mathrm{1}}{\mathrm{2}^{{a}_{{n}} } }\:\:=\:\:\frac{\mathrm{1}}{\mathrm{3}^{{a}_{\mathrm{1}} }…
Question Number 137206 by JulioCesar last updated on 31/Mar/21 Answered by bemath last updated on 31/Mar/21 $$\mathrm{by}\:\mathrm{parts}\:\begin{cases}{\mathrm{u}=\mathrm{ln}\:\left(\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}+\mathrm{1}}\right) \:\mathrm{du}=\frac{\mathrm{2}}{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}+\mathrm{1}\right)}\mathrm{dx}}\\{\mathrm{v}\:=\:\mathrm{x}}\end{cases} \\ $$$$\mathrm{I}\:=\:\mathrm{x}\:\mathrm{ln}\:\left(\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}+\mathrm{1}}\right)−\int\:\frac{\mathrm{2x}}{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}+\mathrm{1}\right)}\mathrm{dx} \\ $$$$\mathrm{I}=\mathrm{x}\:\mathrm{ln}\:\left(\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}+\mathrm{1}}\right)−\left[\int\:\frac{\mathrm{1}}{\mathrm{x}−\mathrm{1}}\mathrm{dx}+\int\:\frac{\mathrm{1}}{\mathrm{x}+\mathrm{1}}\mathrm{dx}\:\right] \\ $$$$\mathrm{I}=\:\mathrm{x}\:\mathrm{ln}\:\left(\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}+\mathrm{1}}\right)−\mathrm{ln}\:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\:+\:\mathrm{C}…
Question Number 6132 by sanusihammed last updated on 15/Jun/16 $${Evaluate}\:{the}\:{integral}\:{of}\:… \\ $$$$ \\ $$$$\left[\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{2}}+\frac{{x}^{\mathrm{5}} }{\mathrm{2}.\mathrm{4}}−\frac{{x}^{\mathrm{7}} }{\mathrm{2}.\mathrm{4}.\mathrm{6}}+….\right)\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} }+\frac{{x}^{\mathrm{4}} }{\mathrm{2}^{\mathrm{2}} .\mathrm{4}^{\mathrm{2}} }−\frac{{x}^{\mathrm{6}} }{\mathrm{2}^{\mathrm{2}} .\mathrm{4}^{\mathrm{2}} .\mathrm{6}^{\mathrm{2}}…
Question Number 71666 by aliesam last updated on 18/Oct/19 $${f}:{z}\rightarrow{z} \\ $$$$ \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right)+\mathrm{3}\left(\mathrm{4}{xy}−\mathrm{1}\right) \\ $$$$ \\ $$$$,{f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$ \\ $$$$\forall{x},{y}\:\in{z} \\ $$$${evaluate}\:{f}\left(\mathrm{19}\right) \\…
Question Number 6131 by love math last updated on 15/Jun/16 $${sin}\:\mathrm{4}{x}+{sin}\mathrm{12}{x}+{sin}\:\mathrm{8}{x} \\ $$ Answered by Irfan hakim last updated on 18/Jun/16 $$\mathrm{sin}\:\mathrm{24}{x} \\ $$ Commented…
Question Number 137203 by mathocean1 last updated on 31/Mar/21 $$\int\:\frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}}=? \\ $$ Answered by Dwaipayan Shikari last updated on 31/Mar/21 $$\int\frac{{log}\left(\mathrm{1}+{x}\right)}{{x}}{dx} \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}}…
Question Number 71664 by mathmax by abdo last updated on 18/Oct/19 $${find}\:{nature}\:{of}\:{the}\:{sequence}\:{U}_{{n}} =\frac{\mathrm{1}}{{n}}\left(\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\right)^{\mathrm{2}} \\ $$ Commented by mathmax by abdo last updated on…
Question Number 71665 by mathmax by abdo last updated on 18/Oct/19 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left(\sqrt{{x}^{\mathrm{2}} +\mathrm{4}}\right)} \\ $$ Answered by MJS last updated on 19/Oct/19…