Question Number 65569 by MJS last updated on 31/Jul/19 $$\mathrm{reposting}\:\mathrm{this}: \\ $$$${x}^{\mathrm{4}} +\left(\mathrm{1}−\mathrm{2}{a}\right){x}^{\mathrm{2}} −\mathrm{2}{ax}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{solve}\:\mathrm{for}\:{a}>\mathrm{0}\wedge{x}>\mathrm{0} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{5}} −\frac{\mathrm{23}}{\mathrm{8}}{x}^{\mathrm{4}} −\frac{\mathrm{49}}{\mathrm{4}}{x}^{\mathrm{3}} +\frac{\mathrm{27}}{\mathrm{8}}{x}^{\mathrm{2}} +\mathrm{3}{x}+\frac{\mathrm{9}}{\mathrm{8}}=\mathrm{0} \\…
Question Number 31 by user2 last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\int\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}{dx}. \\ $$ Answered by user2 last updated on 03/Nov/14 $$\int\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx}=\int\sqrt{\mathrm{sin}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}+\mathrm{cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}+\mathrm{2sin}\frac{{x}}{\mathrm{2}}\mathrm{cos}\frac{{x}}{\mathrm{2}}\:}{dx} \\ $$$$=\int\left(\mathrm{sin}\frac{{x}}{\mathrm{2}}+\mathrm{cos}\frac{{x}}{\mathrm{2}}\right){dx} \\…
Question Number 131103 by mnjuly1970 last updated on 01/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:{prove}\:\:{that}\::: \\ $$$$\:\:\underset{{n}\in\mathbb{N}} {\sum}\left(\frac{\Gamma^{\mathrm{2}} \left({n}\right)}{\mathrm{2}^{−{n}} \left(\mathrm{2}{n}−\mathrm{1}\right)!}\right)=\pi \\ $$$$ \\ $$ Answered by Ar Brandon…
Question Number 29 by user1 last updated on 25/Jan/15 $$\mathrm{Differentiate}\:\:\:{e}^{\sqrt{\mathrm{cot}\:{x}}} . \\ $$ Answered by user2 last updated on 03/Nov/14 $$\mathrm{Let}\:\:\:{y}={e}^{\sqrt{\mathrm{cot}\:{x}}} \\ $$$$\mathrm{Put}\:\mathrm{cot}\:{x}\:={t}\:\mathrm{and}\:\sqrt{\mathrm{cot}\:{x}}=\sqrt{{t}}={u},\:\mathrm{so}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}={e}^{{u}}…
Question Number 131097 by dw last updated on 01/Feb/21 $${Determine}\:{the}\:{value}\:{of}\:\frac{{S}}{\pi},\:{if}\:{S}\:{is}\:{the}\:{sum},\:{in}\:{radians}, \\ $$$${all}\:{equation}\:{solutions}\:{contained}\:{in}\:{the}\:{interval}\:\left[\mathrm{0},\mathrm{14}\pi\right]. \\ $$$$ \\ $$$${the}\:{equation}\:{is}: \\ $$$${cos}\left({x}\right)+{cos}^{\mathrm{5}} \left({x}\right)+{cos}\left(\mathrm{7}{x}\right)=\mathrm{3} \\ $$ Answered by MJS_new last…
Question Number 27 by user1 last updated on 25/Jan/15 $$\mathrm{If}\:\boldsymbol{\mathrm{f}}={x}^{\mathrm{2}} {z}\boldsymbol{\mathrm{i}}−\mathrm{2}{y}^{\mathrm{3}} {z}^{\mathrm{2}} \boldsymbol{\mathrm{j}}+{xy}^{\mathrm{2}} {z}\boldsymbol{\mathrm{k}}.\:\mathrm{Find}\:{div}\:\boldsymbol{\mathrm{f}},\:{curl}\:\boldsymbol{\mathrm{f}},\: \\ $$$${at}\left(\mathrm{1},\:−\mathrm{1},\:\mathrm{1}\right). \\ $$ Answered by user1 last updated on 03/Nov/14…
Question Number 131099 by mohammad17 last updated on 01/Feb/21 Commented by mohammad17 last updated on 01/Feb/21 $${how}\:{can}\:{solve}\:{this} \\ $$$$ \\ $$ Answered by mr W…
Question Number 25 by user1 last updated on 25/Jan/15 $$\mathrm{Expand}\:\:\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)^{\mathrm{2}} \\ $$ Answered by user1 last updated on 31/Oct/14 $$\:\left({a}+{b}+{c}\right)\left({a}+{b}+{c}\right) \\ $$$$={a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{2}{bc}+\mathrm{2}{ca}…
Question Number 23 by user1 last updated on 25/Jan/15 $$\mathrm{Sum}\:\mathrm{the}\:\mathrm{series}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\boldsymbol{\mathrm{P}}_{{r}} \left({n}\right)\frac{{x}^{{n}} }{{n}!}\:,\:\mathrm{where}\:\boldsymbol{\mathrm{P}}_{{r}} \left({n}\right)\: \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{of}\:\mathrm{degree}\:{r}\:\mathrm{in}\:{n}. \\ $$ Answered by user1 last updated on…
Question Number 21 by user1 last updated on 25/Jan/15 $$\mathrm{A}\:\mathrm{card}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\mathrm{a}\:\mathrm{well} \\ $$$$\mathrm{shuffled}\:\mathrm{deck}\:\mathrm{of}\:\mathrm{52}\:\mathrm{cards}.\:\mathrm{Find}\:\mathrm{the}\:\: \\ $$$$\mathrm{probability}\:\mathrm{of}\:\mathrm{its}\:\mathrm{being}\:\mathrm{a}\:\mathrm{spade}\:\mathrm{or}\:\mathrm{a}\:\mathrm{king}. \\ $$ Answered by user1 last updated on 30/Oct/14 $$\mathrm{Let}\:{S}\:\mathrm{be}\:\mathrm{the}\:\mathrm{sample}\:\mathrm{space}.\:{n}\left({S}\right)=\mathrm{52}.\: \\…