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Question Number 100866 Answers: 1 Comments: 0

solve 3x^2 y^(′′) −2xy^′ +4y =0

$$\mathrm{solve}\:\mathrm{3x}^{\mathrm{2}} \mathrm{y}^{''} −\mathrm{2xy}^{'} \:+\mathrm{4y}\:=\mathrm{0} \\ $$

Question Number 100948 Answers: 0 Comments: 1

Question Number 100843 Answers: 0 Comments: 0

Σ_(k=1) ^n ((ln(k))/(k!)) = ?

$$\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\boldsymbol{{ln}}\left(\boldsymbol{{k}}\right)}{\boldsymbol{{k}}!}\:=\:? \\ $$

Question Number 100850 Answers: 2 Comments: 1

∫_0 ^(102) (x−1)(x−2).....(x−100)×((1/(x−1))+(1/(x−2))+...+(1/(x−100)))dx

$$\int_{\mathrm{0}} ^{\mathrm{102}} \left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right).....\left({x}−\mathrm{100}\right)×\left(\frac{\mathrm{1}}{{x}−\mathrm{1}}+\frac{\mathrm{1}}{{x}−\mathrm{2}}+...+\frac{\mathrm{1}}{{x}−\mathrm{100}}\right){dx} \\ $$

Question Number 100832 Answers: 1 Comments: 3

log(√(125)) ∙ln10 ∙log_5 e=? help me

$$\mathrm{log}\sqrt{\mathrm{125}}\:\centerdot\mathrm{ln10}\:\centerdot\mathrm{log}_{\mathrm{5}} \mathrm{e}=? \\ $$$$\mathrm{help}\:\mathrm{me} \\ $$

Question Number 100829 Answers: 0 Comments: 0

hello every one prove that ∫_0 ^(π/2) cos^u (x) cos(ax) arctan(b cos(x)) dx =((2^(−u−2) .π.b.Γ(u+2))/(Γ(((u−a+3)/2))Γ(((u+a+3)/2)))).x_4 F_3 ((((1/2),1+(u/2),((u+3)/2),−b^2 )),(((3/2),((u−a+3)/2),((u+a+3)/2))) ) Re u>−1 ,∣arg(1+b^2 ) ∣<π

$${hello}\:{every}\:{one}\: \\ $$$$ \\ $$$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cos}^{{u}} \left({x}\right)\:{cos}\left({ax}\right)\:{arctan}\left({b}\:{cos}\left({x}\right)\right)\:{dx} \\ $$$$=\frac{\mathrm{2}^{−{u}−\mathrm{2}} .\pi.{b}.\Gamma\left({u}+\mathrm{2}\right)}{\Gamma\left(\frac{{u}−{a}+\mathrm{3}}{\mathrm{2}}\right)\Gamma\left(\frac{{u}+{a}+\mathrm{3}}{\mathrm{2}}\right)}.{x}_{\mathrm{4}} {F}_{\mathrm{3}} \begin{pmatrix}{\frac{\mathrm{1}}{\mathrm{2}},\mathrm{1}+\frac{{u}}{\mathrm{2}},\frac{{u}+\mathrm{3}}{\mathrm{2}},−{b}^{\mathrm{2}} }\\{\frac{\mathrm{3}}{\mathrm{2}},\frac{{u}−{a}+\mathrm{3}}{\mathrm{2}},\frac{{u}+{a}+\mathrm{3}}{\mathrm{2}}}\end{pmatrix} \\ $$$$ \\ $$$$ \\ $$$${Re}\:{u}>−\mathrm{1}\:,\mid{arg}\left(\mathrm{1}+{b}^{\mathrm{2}} \right)\:\mid<\pi \\ $$$$ \\ $$

Question Number 100817 Answers: 2 Comments: 1

Question Number 100815 Answers: 1 Comments: 0

Question Number 100806 Answers: 1 Comments: 3

Question Number 100793 Answers: 1 Comments: 1

Question Number 100792 Answers: 1 Comments: 0

Question Number 100789 Answers: 2 Comments: 0

Question Number 100785 Answers: 1 Comments: 0

Question Number 100771 Answers: 2 Comments: 1

sho that (0,(1/2)) is a point of symetry for the curve f(x) = x +(1/(1−e^x )) Please make a reference to a book i can understand centre of symmetry of rational functions and functions like this

$$\:\mathrm{sho}\:\mathrm{that}\:\:\left(\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{of}\:\mathrm{symetry}\:\mathrm{for}\:\mathrm{the}\:\mathrm{curve} \\ $$$$\:{f}\left({x}\right)\:=\:{x}\:+\frac{\mathrm{1}}{\mathrm{1}−{e}^{{x}} } \\ $$$$\mathrm{Please}\:\mathrm{make}\:\mathrm{a}\:\mathrm{reference}\:\mathrm{to}\:\mathrm{a}\:\mathrm{book}\:\mathrm{i}\:\mathrm{can}\:\mathrm{understand} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{symmetry}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{functions}\:\mathrm{and}\:\mathrm{functions} \\ $$$$\mathrm{like}\:\mathrm{this} \\ $$

Question Number 100769 Answers: 0 Comments: 0

Consider the sequences (u_n ) and (v_n ) defined by { ((u_0 = 1)),((u_(n+1) = ((2u_n v_n )/(u_n + v_n )))) :} and { ((v_0 = 2)),((v_(n+1) = ((u_n + v_n )/2))) :} ∀ n∈ N (1) Show that (u_n ) and (v_n ) are strictly positive also Show that (u_n ) and (v_n ) are of opposite sense of variation. (2) let w_n = v_n −u_n show that 0 ≤ w_(n+1) ≤ (1/2)w_n (3) Prove by induction that 0 ≤ w_n ≤ (1/2^n )

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{sequences}\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \right)\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:\begin{cases}{{u}_{\mathrm{0}} \:=\:\mathrm{1}}\\{{u}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{2}{u}_{{n}} {v}_{{n}} }{{u}_{{n}} \:+\:{v}_{{n}} }}\end{cases}\:\mathrm{and}\:\begin{cases}{{v}_{\mathrm{0}} \:=\:\mathrm{2}}\\{{v}_{{n}+\mathrm{1}} \:=\:\frac{{u}_{{n}} \:+\:{v}_{{n}} }{\mathrm{2}}}\end{cases}\:\:\forall\:{n}\in\:\mathbb{N} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Show}\:\mathrm{that}\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \right)\:\mathrm{are}\:\mathrm{strictly}\:\mathrm{positive}\:\mathrm{also} \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \:\right)\:\mathrm{are}\:\mathrm{of}\:\mathrm{opposite}\:\mathrm{sense}\:\mathrm{of}\:\mathrm{variation}. \\ $$$$\left(\mathrm{2}\right)\:\mathrm{let}\:{w}_{{n}} \:=\:{v}_{{n}} −{u}_{{n}} \:\:\mathrm{show}\:\mathrm{that}\:\:\mathrm{0}\:\leqslant\:{w}_{{n}+\mathrm{1}} \:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}}{w}_{{n}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Prove}\:\mathrm{by}\:\mathrm{induction}\:\mathrm{that}\:\mathrm{0}\:\leqslant\:{w}_{{n}} \:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}^{{n}} } \\ $$

Question Number 100767 Answers: 2 Comments: 0

sketch x^2 = y^3 and x^2 + y^2 = 16 and hence solve (x^2 −y^3 )(x^2 + y^2 −16) ≥0

$$\:\mathrm{sketch}\:\:{x}^{\mathrm{2}} \:=\:{y}^{\mathrm{3}} \:\mathrm{and}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:=\:\mathrm{16} \\ $$$$\mathrm{and}\:\mathrm{hence}\:\mathrm{solve}\:\left({x}^{\mathrm{2}} −{y}^{\mathrm{3}} \right)\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} −\mathrm{16}\right)\:\geqslant\mathrm{0} \\ $$

Question Number 100766 Answers: 4 Comments: 0

Question Number 100746 Answers: 1 Comments: 0

∫_(−∞) ^∞ ((cos3x)/((1+x^2 )^2 ))dx

$$\int_{−\infty} ^{\infty} \frac{{cos}\mathrm{3}{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 100744 Answers: 0 Comments: 17

Happy tau day to all !!! 28 june

$$\mathrm{Happy}\:\boldsymbol{\mathrm{tau}}\:\boldsymbol{\mathrm{day}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{all}}\:!!! \\ $$$$ \\ $$$$\mathrm{28}\:{june}\: \\ $$

Question Number 100743 Answers: 1 Comments: 0

(√(((2+5)/(35))+6))+30×((15+90)/(255))

$$\sqrt{\frac{\mathrm{2}+\mathrm{5}}{\mathrm{35}}+\mathrm{6}}+\mathrm{30}×\frac{\mathrm{15}+\mathrm{90}}{\mathrm{255}} \\ $$

Question Number 100733 Answers: 0 Comments: 6

A positive integer such as 4334 is a palindrome if it reads the same forwards or backwards. What is the only prime palindrome with an even number of digits?

$$\mathrm{A}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such}\:\mathrm{as}\:\mathrm{4334}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{palindrome}\:\mathrm{if}\:\mathrm{it}\:\mathrm{reads}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{forwards}\:\mathrm{or}\:\mathrm{backwards}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{only}\:\mathrm{prime}\:\mathrm{palindrome}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{even}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digits}?\: \\ $$

Question Number 100732 Answers: 1 Comments: 5

(1) If ((x+yi)/(1+i)) = (7/(7+i)) where x and y are real , what is the value of x+y (2)What are all values of x which satisfy x^2 −cos x+1 = 0 (3)What are all values of x between 0^o and 360^o which satisfy (5+2(√6))^(sin x) + (5−2(√6))^(sin x) = 2(√3)

$$\left(\mathrm{1}\right)\:\mathrm{If}\:\frac{{x}+{yi}}{\mathrm{1}+{i}}\:=\:\frac{\mathrm{7}}{\mathrm{7}+{i}}\:\mathrm{where}\:{x}\:\mathrm{and}\:\mathrm{y}\: \\ $$$$\mathrm{are}\:\mathrm{real}\:,\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}+{y}\: \\ $$$$\left(\mathrm{2}\right)\mathrm{What}\:\mathrm{are}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{which} \\ $$$$\mathrm{satisfy}\:{x}^{\mathrm{2}} −\mathrm{cos}\:{x}+\mathrm{1}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{3}\right){W}\mathrm{hat}\:\mathrm{are}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{between}\: \\ $$$$\mathrm{0}^{\mathrm{o}} \:\mathrm{and}\:\mathrm{360}^{\mathrm{o}} \:\mathrm{which}\:\mathrm{satisfy}\: \\ $$$$\left(\mathrm{5}+\mathrm{2}\sqrt{\mathrm{6}}\right)^{\mathrm{sin}\:\mathrm{x}} \:+\:\left(\mathrm{5}−\mathrm{2}\sqrt{\mathrm{6}}\right)^{\mathrm{sin}\:\mathrm{x}} \:=\:\mathrm{2}\sqrt{\mathrm{3}} \\ $$

Question Number 100730 Answers: 0 Comments: 1

((2z^2 )/(z^2 +∣z+1∣)) < 1

$$\frac{\mathrm{2}{z}^{\mathrm{2}} }{{z}^{\mathrm{2}} +\mid{z}+\mathrm{1}\mid}\:<\:\mathrm{1}\: \\ $$

Question Number 100720 Answers: 1 Comments: 3

Question Number 100703 Answers: 0 Comments: 2

Question Number 100696 Answers: 0 Comments: 4

1. 25≠66 = True or False? 2. 44=44 =True or false? 3. 39>169=True or False? 4. 15<61 = True or false? Make sure you have to answer correctly

$$\mathrm{1}.\:\:\:\:\mathrm{25}\neq\mathrm{66}\:=\:\mathrm{True}\:\mathrm{or}\:\mathrm{False}? \\ $$$$\mathrm{2}.\:\:\:\:\mathrm{44}=\mathrm{44}\:=\mathrm{True}\:\mathrm{or}\:\mathrm{false}? \\ $$$$\mathrm{3}.\:\:\:\:\:\mathrm{39}>\mathrm{169}=\mathrm{True}\:\mathrm{or}\:\mathrm{False}? \\ $$$$\mathrm{4}.\:\:\:\:\mathrm{15}<\mathrm{61}\:=\:\mathrm{True}\:\mathrm{or}\:\mathrm{false}? \\ $$$$\: \\ $$$$\mathrm{Make}\:\mathrm{sure}\:\mathrm{you}\:\mathrm{have}\:\mathrm{to} \\ $$$$\mathrm{answer}\:\mathrm{correctly} \\ $$

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