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Question Number 86791    Answers: 4   Comments: 0

Solve the equation: x ≡ 3 (mod 5) x ≡ 4 (mod 7) x ≡ 2 (mod 3)

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\mathrm{x}\:\:\equiv\:\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{5}\right) \\ $$$$\:\:\:\:\mathrm{x}\:\:\equiv\:\:\mathrm{4}\:\left(\mathrm{mod}\:\mathrm{7}\right) \\ $$$$\:\:\:\:\mathrm{x}\:\:\equiv\:\:\mathrm{2}\:\left(\mathrm{mod}\:\mathrm{3}\right) \\ $$

Question Number 86789    Answers: 2   Comments: 1

Question Number 86786    Answers: 0   Comments: 6

show that Σ_(n=2) ^∞ (1/(n^2 (1−n^2 )^2 ))=0.2999

$${show}\:{that} \\ $$$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left(\mathrm{1}−{n}^{\mathrm{2}} \right)^{\mathrm{2}} }=\mathrm{0}.\mathrm{2999} \\ $$

Question Number 86779    Answers: 1   Comments: 0

ssolve 1)x−[x]≥0 2)x−[x]≤0 3)x+[x]≥0 4)x+[x]≤0

$${ssolve} \\ $$$$\left.\mathrm{1}\right){x}−\left[{x}\right]\geqslant\mathrm{0} \\ $$$$\left.\mathrm{2}\right){x}−\left[{x}\right]\leqslant\mathrm{0} \\ $$$$\left.\mathrm{3}\right){x}+\left[{x}\right]\geqslant\mathrm{0} \\ $$$$\left.\mathrm{4}\right){x}+\left[{x}\right]\leqslant\mathrm{0}\: \\ $$

Question Number 86762    Answers: 0   Comments: 4

Find the sum of the series 1+(1/2)+(1/3)+(1/4)+(1/6)+(1/8)+(1/9)+(1/(12))+∙∙∙ where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s

$${Find}\:{the}\:{sum}\:{of}\:{the}\:{series} \\ $$$$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{8}}+\frac{\mathrm{1}}{\mathrm{9}}+\frac{\mathrm{1}}{\mathrm{12}}+\centerdot\centerdot\centerdot \\ $$$${where}\:{the}\:{terms}\:{are}\:{the}\:{reciprocals} \\ $$$${of}\:{the}\:{positive}\:{integers}\:{whose}\:{only}\: \\ $$$${prime}\:{factors}\:{are}\:\mathrm{2}{s}\:{and}\:\mathrm{3}{s} \\ $$

Question Number 86761    Answers: 0   Comments: 0

∫ ((x+ sin x)/(x+ cos x)) dx =

$$\int\:\:\frac{\mathrm{x}+\:\mathrm{sin}\:\mathrm{x}}{\mathrm{x}+\:\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:=\: \\ $$

Question Number 86741    Answers: 1   Comments: 4

{ ((x+10y+50z=500)),((x+y+z=100)) :} find x,y,z

$$\begin{cases}{{x}+\mathrm{10}{y}+\mathrm{50}{z}=\mathrm{500}}\\{{x}+{y}+{z}=\mathrm{100}}\end{cases} \\ $$$$ \\ $$$${find}\:{x},{y},{z} \\ $$

Question Number 86737    Answers: 2   Comments: 4

prove that 1/cos2x+cosx+1=((sin((5x)/2))/(2sin(x/2)))+(1/2) 2/((cos(x)+isin(x)−1)/(cos(x)+isin(x)+1))=−i tan(x) 3/((cos(5x)+isin(5x)+1)/(cos(5x)−isin(x)+1))=cos(5x)+isin(5x)

$${prove}\:{that} \\ $$$$\mathrm{1}/{cos}\mathrm{2}{x}+{cosx}+\mathrm{1}=\frac{{sin}\frac{\mathrm{5}{x}}{\mathrm{2}}}{\mathrm{2}{sin}\frac{{x}}{\mathrm{2}}}+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{2}/\frac{{cos}\left({x}\right)+{isin}\left({x}\right)−\mathrm{1}}{{cos}\left({x}\right)+{isin}\left({x}\right)+\mathrm{1}}=−{i}\:{tan}\left({x}\right) \\ $$$$ \\ $$$$\mathrm{3}/\frac{{cos}\left(\mathrm{5}{x}\right)+{isin}\left(\mathrm{5}{x}\right)+\mathrm{1}}{{cos}\left(\mathrm{5}{x}\right)−{isin}\left({x}\right)+\mathrm{1}}={cos}\left(\mathrm{5}{x}\right)+{isin}\left(\mathrm{5}{x}\right) \\ $$

Question Number 86734    Answers: 1   Comments: 0

Find all functions that satisfy the equation [∫f(x)dx][∫(1/(f(x)))dx]=−1

$${Find}\:{all}\:{functions}\:{that}\:{satisfy}\:{the} \\ $$$${equation} \\ $$$$\left[\int{f}\left({x}\right){dx}\right]\left[\int\frac{\mathrm{1}}{{f}\left({x}\right)}{dx}\right]=−\mathrm{1} \\ $$

Question Number 86728    Answers: 0   Comments: 1

∫_0 ^∞ ln(1+(b^2 /x^2 )) dx

$$\int_{\mathrm{0}} ^{\infty} {ln}\left(\mathrm{1}+\frac{{b}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right)\:{dx} \\ $$

Question Number 86726    Answers: 0   Comments: 0

Question Number 86723    Answers: 0   Comments: 1

Question Number 86716    Answers: 2   Comments: 2

cos ((π/(17)))×cos (((2π)/(17)))×cos (((4π)/(17)))×cos (((8π)/(17))) =

$$\mathrm{cos}\:\left(\frac{\pi}{\mathrm{17}}\right)×\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{17}}\right)×\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{17}}\right)×\mathrm{cos}\:\left(\frac{\mathrm{8}\pi}{\mathrm{17}}\right)\:= \\ $$

Question Number 86708    Answers: 1   Comments: 0

∫x (√((√2) x−(√(2x^2 −1)))) dx

$$\int\mathrm{x}\:\sqrt{\sqrt{\mathrm{2}}\:\mathrm{x}−\sqrt{\mathrm{2x}^{\mathrm{2}} −\mathrm{1}}}\:\mathrm{dx}\: \\ $$

Question Number 86703    Answers: 3   Comments: 6

I=∫(1/(x^4 +1))dx

$${I}=\int\frac{\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{1}}{dx} \\ $$

Question Number 86701    Answers: 0   Comments: 1

find solution 4^(sin x −(1/4)) − (1/(2+(√2))) .2^(sin x) −1 = 0 in x ∈[ 0,2π ]

$$\mathrm{find}\:\mathrm{solution}\: \\ $$$$\mathrm{4}^{\mathrm{sin}\:\mathrm{x}\:−\frac{\mathrm{1}}{\mathrm{4}}} \:−\:\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{2}}}\:.\mathrm{2}^{\mathrm{sin}\:\mathrm{x}} \:−\mathrm{1}\:=\:\mathrm{0}\: \\ $$$$\mathrm{in}\:\mathrm{x}\:\in\left[\:\mathrm{0},\mathrm{2}\pi\:\right]\: \\ $$

Question Number 86684    Answers: 2   Comments: 0

If (1+px+qx^2 )^8 = 1+8x+52x^2 +kx^3 +... find p , q and k.

$$\mathrm{If}\:\left(\mathrm{1}+\mathrm{px}+\mathrm{qx}^{\mathrm{2}} \right)^{\mathrm{8}} \:=\:\mathrm{1}+\mathrm{8x}+\mathrm{52x}^{\mathrm{2}} +\mathrm{kx}^{\mathrm{3}} +... \\ $$$$\mathrm{find}\:\mathrm{p}\:,\:\mathrm{q}\:\mathrm{and}\:\mathrm{k}.\: \\ $$

Question Number 86675    Answers: 1   Comments: 7

lim_(x→∞) ((∫_0 ^1 (1+x^n )^n dx))^(1/n) =?

$$\underset{{x}\rightarrow\infty} {{lim}}\sqrt[{{n}}]{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}^{{n}} \right)^{{n}} {dx}}=? \\ $$

Question Number 86672    Answers: 0   Comments: 0

show proofs by induction,that ((x_1 +x_2 +....+x_n )/n)≥(x_1 x_2 ....x_n )^(1/n) ∀n=2^k ,k>1 and (x_1 ,x_2 ,x_3 ,.....x_n )>0.

$${show}\:{proofs}\:{by}\:{induction},{that} \\ $$$$\frac{{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +....+{x}_{{n}} }{{n}}\geqslant\left({x}_{\mathrm{1}} {x}_{\mathrm{2}} ....{x}_{{n}} \right)^{\frac{\mathrm{1}}{{n}}} \\ $$$$\forall{n}=\mathrm{2}^{{k}} ,{k}>\mathrm{1}\:{and}\:\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,{x}_{\mathrm{3}} ,.....{x}_{{n}} \right)>\mathrm{0}. \\ $$

Question Number 86671    Answers: 1   Comments: 0

∫sin(x) arcsin(x)

$$\int{sin}\left({x}\right)\:{arcsin}\left({x}\right) \\ $$

Question Number 86668    Answers: 0   Comments: 1

what is P(∣x∣ > 1 ) if x has a PDF of f(x) = { (((1/4) , −2<x<2)),((0 , elsewhere)) :}

$$\mathrm{what}\:\mathrm{is}\:\mathrm{P}\left(\mid\mathrm{x}\mid\:>\:\mathrm{1}\:\right)\:\mathrm{if}\:\mathrm{x}\:\mathrm{has}\:\mathrm{a}\:\mathrm{PDF}\:\mathrm{of} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\begin{cases}{\frac{\mathrm{1}}{\mathrm{4}}\:,\:\:−\mathrm{2}<\mathrm{x}<\mathrm{2}}\\{\mathrm{0}\:,\:\mathrm{elsewhere}}\end{cases} \\ $$

Question Number 86663    Answers: 1   Comments: 1

The matrix A satisfying the equation [(1,3),(0,1) ]A = [(1,( 1)),(0,(−1)) ] is

$$\mathrm{The}\:\mathrm{matrix}\:{A}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{3}}\\{\mathrm{0}}&{\mathrm{1}}\end{bmatrix}{A}\:=\:\begin{bmatrix}{\mathrm{1}}&{\:\:\:\:\mathrm{1}}\\{\mathrm{0}}&{−\mathrm{1}}\end{bmatrix}\:\mathrm{is} \\ $$

Question Number 86657    Answers: 1   Comments: 0

solve (1+x^3 )dy −x^2 y dx=0 y(1) = 2

$$\mathrm{solve}\:\left(\mathrm{1}+\mathrm{x}^{\mathrm{3}} \right)\mathrm{dy}\:−\mathrm{x}^{\mathrm{2}} \:\mathrm{y}\:\mathrm{dx}=\mathrm{0} \\ $$$$\mathrm{y}\left(\mathrm{1}\right)\:=\:\mathrm{2} \\ $$

Question Number 86655    Answers: 1   Comments: 0

lim_(x→−1^+ ) (((√π) −(√(arc cos x)))/(√(x+1)))

$$\underset{{x}\rightarrow−\mathrm{1}^{+} } {\mathrm{lim}}\:\frac{\sqrt{\pi}\:−\sqrt{\mathrm{arc}\:\mathrm{cos}\:\mathrm{x}}}{\sqrt{\mathrm{x}+\mathrm{1}}} \\ $$

Question Number 86646    Answers: 0   Comments: 2

∫_0 ^1 ((ln(1+x))/(x^2 +1))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 86643    Answers: 0   Comments: 2

calculate ∫_0 ^∞ ((cos(2ch(x)))/(x^2 +9))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{ch}\left({x}\right)\right)}{{x}^{\mathrm{2}} \:+\mathrm{9}}{dx} \\ $$

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