Question Number 597 by 112358 last updated on 08/Feb/15 $${Evaluate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\mathrm{1}+{u}^{\mathrm{4}} }{du}\:{exactly}\:{if}\: \\ $$$${u}={coshx}.\: \\ $$ Commented by prakash jain last updated on 08/Feb/15…
Question Number 596 by 123456 last updated on 08/Feb/15 $$\int\underset{{s}} {\int}\frac{{dx}\wedge{dy}+{dx}\wedge{dz}−{dy}\wedge{dz}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} } \\ $$$${where}\:{s}\:{is}\:{the}\:{surface}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{1}\: \\ $$ Terms of Service Privacy…
Question Number 131665 by naka3546 last updated on 07/Feb/21 Answered by mr W last updated on 07/Feb/21 Commented by mr W last updated on 07/Feb/21…
Question Number 593 by jitendrarathod2556@gmail.com last updated on 06/Feb/15 $$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$ Commented by 123456 last updated…
Question Number 131661 by benjo_mathlover last updated on 07/Feb/21 $$\:\:\:\mathrm{log}\:_{\sqrt{\mathrm{2}}\:\mathrm{sin}\:\mathrm{x}} \left(\mathrm{1}+\mathrm{cos}\:\mathrm{x}\right)\:=\:\mathrm{2} \\ $$$$\:−\frac{\mathrm{2}\pi}{\mathrm{3}}\leqslant\mathrm{x}\leqslant\frac{\pi}{\mathrm{3}} \\ $$ Answered by liberty last updated on 07/Feb/21 $$\:\begin{cases}{\sqrt{\mathrm{2}}\:\mathrm{sin}\:\mathrm{x}\:>\mathrm{0}\:;\:\mathrm{x}\:\mathrm{in}\:\mathrm{I}\:\mathrm{or}\:\mathrm{II}\:\mathrm{quadrant}}\\{\sqrt{\mathrm{2}}\:\mathrm{sin}\:\mathrm{x}\:\neq\:\mathrm{1}\Rightarrow\mathrm{sin}\:\mathrm{x}\neq\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}\end{cases} \\ $$$$\Leftrightarrow\:\mathrm{1}+\mathrm{cos}\:\mathrm{x}\:=\:\left(\sqrt{\mathrm{2}}\:\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{2}}…
Question Number 66126 by Joel122 last updated on 09/Aug/19 $$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{formula}\:\mathrm{to}\:\mathrm{find}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{1}\:+\:{n}^{\mathrm{2}} \:+\:{n}^{\mathrm{4}} \:+\:{n}^{\mathrm{6}} \:+\:{n}^{\mathrm{8}} \:+\:…\:+\:{n}^{\mathrm{2}{k}} \:+\:… \\ $$$$\mathrm{where}\:{n},{k}\:\in\:\mathbb{Z}^{+} \: \\ $$ Answered by mr…
Question Number 591 by 112358 last updated on 04/Feb/15 $${Prove}\:{by}\:{induction}\:{on}\:{n},\:{for}\:{n}\geqslant\mathrm{2}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{u}_{{n}} \:\geqslant\:\mathrm{2}^{\mathrm{3}^{{n}−\mathrm{1}} } \\ $$$${for}\:{the}\:{sequence}\:\left\{{u}_{{n}} \right\}\:{defined}\:{by}\: \\ $$$${the}\:{recurrence}\:{relation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{u}_{\mathrm{1}} =\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{u}_{{n}+\mathrm{1}} =\left({u}_{{n}}…
Question Number 588 by shaleen last updated on 03/Feb/15 $$\left({a}+{b}=\mathrm{10}\right)\:\left({a}+{c}=\mathrm{24}\right)\:\left({b}+{c}=\mathrm{20}\right) \\ $$$${find}\:{a}\:{b}\:{c} \\ $$ Answered by prakash jain last updated on 03/Feb/15 $${a}+{b}=\mathrm{10}\Rightarrow{b}=\mathrm{10}−{a} \\ $$$${a}+{c}=\mathrm{24}\Rightarrow{c}=\mathrm{24}−{a}…
Question Number 585 by 123456 last updated on 02/Feb/15 $${f}_{{n}} \left({x}\right)=\left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{nx}} ,{n}\in\mathbb{N}^{\ast} ,{x}\in\mathbb{R},{x}\geqslant\mathrm{0} \\ $$$$\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\:{f}_{{n}} \left({x}\right)= \\ $$ Answered by prakash jain last updated…
Question Number 582 by ssahoo last updated on 31/Jan/15 $$\mathrm{Show}\:\mathrm{that} \\ $$$$\sqrt{\mathrm{5}+\sqrt{\mathrm{21}}}\:+\sqrt{\mathrm{8}+\sqrt{\mathrm{55}}}\:=\sqrt{\mathrm{7}+\sqrt{\mathrm{33}}}\:+\sqrt{\mathrm{6}+\sqrt{\mathrm{35}}} \\ $$ Commented by prakash jain last updated on 31/Jan/15 $$\sqrt{\mathrm{5}+\sqrt{\mathrm{21}}}={a}+{b} \\ $$$${a}^{\mathrm{2}}…