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Question Number 159737 Answers: 0 Comments: 0

Prove 1) E(x)+E(y)≤E(x+y)≤E(x)+E(y)+1 2) E(x)+E(y)+E(x+1)≤E(2x)+E(2y) 3) E((x/2))+E(((x+1)/2))=E(x)

$${Prove}\: \\ $$$$\left.\mathrm{1}\right)\:{E}\left({x}\right)+{E}\left({y}\right)\leqslant{E}\left({x}+{y}\right)\leqslant{E}\left({x}\right)+{E}\left({y}\right)+\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{E}\left({x}\right)+{E}\left({y}\right)+{E}\left({x}+\mathrm{1}\right)\leqslant{E}\left(\mathrm{2}{x}\right)+{E}\left(\mathrm{2}{y}\right) \\ $$$$\left.\mathrm{3}\right)\:{E}\left(\frac{{x}}{\mathrm{2}}\right)+{E}\left(\frac{{x}+\mathrm{1}}{\mathrm{2}}\right)={E}\left({x}\right) \\ $$

Question Number 159736 Answers: 0 Comments: 0

Prove that 1)Sup(A∪B)=max(Sup(A), Sup(B)) 2) inf(A∪B)=min(inf(A), inf(B))

$${Prove}\:{that} \\ $$$$\left.\mathrm{1}\right){Sup}\left({A}\cup{B}\right)={ma}\mathrm{x}\left(\mathrm{S}{up}\left({A}\right),\:{Sup}\left({B}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{inf}\left({A}\cup{B}\right)={min}\left({inf}\left({A}\right),\:{inf}\left({B}\right)\right) \\ $$

Question Number 159733 Answers: 0 Comments: 0

lim_(x→0^+ ) ((⌊xcos(x)⌋)/(xsin(π⌊(e^(1/x) /(ln(x)))⌋)))

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\lfloor{xcos}\left({x}\right)\rfloor}{{xsin}\left(\pi\lfloor\frac{{e}^{\frac{\mathrm{1}}{{x}}} }{{ln}\left({x}\right)}\rfloor\right)} \\ $$

Question Number 159731 Answers: 0 Comments: 0

Question Number 159730 Answers: 0 Comments: 0

Question Number 159727 Answers: 1 Comments: 0

lim_(x→0) ((1+tan (1−((x/(sin x))))))^(1/x^3 ) ?

$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt[{{x}^{\mathrm{3}} }]{\mathrm{1}+\mathrm{tan}\:\left(\mathrm{1}−\left(\frac{{x}}{\mathrm{sin}\:{x}}\right)\right)}\:?\: \\ $$

Question Number 159725 Answers: 0 Comments: 0

Question Number 159724 Answers: 0 Comments: 0

Question Number 159723 Answers: 1 Comments: 1

(1/(k+1))≤∫_k ^(k+1) ((1/x))dx≤(1/k) please show it with k∈ℵ−(0)

$$\frac{\mathrm{1}}{{k}+\mathrm{1}}\leqslant\int_{{k}} ^{{k}+\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right){dx}\leqslant\frac{\mathrm{1}}{{k}}\:\:\:\:\:\:\:\:\: \\ $$$$\:{please}\:{show}\:{it}\:{with}\:{k}\in\aleph−\left(\mathrm{0}\right) \\ $$$$ \\ $$

Question Number 159720 Answers: 1 Comments: 1

L = lim_(x→(π/3)) ((3−4sin^2 x)/(sin 2x−sin x)) ? Q = lim_(x→0) [(1/x^2 ) ((2/(cos^2 x)) +cos x−3)] ?

$$\:\:\:\:\:\:\:\:{L}\:=\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{3}}} {\mathrm{lim}}\:\frac{\mathrm{3}−\mathrm{4sin}\:^{\mathrm{2}} {x}}{\mathrm{sin}\:\mathrm{2}{x}−\mathrm{sin}\:{x}}\:? \\ $$$$\:\:\:\:\:\:{Q}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:\left(\frac{\mathrm{2}}{\mathrm{cos}\:^{\mathrm{2}} {x}}\:+\mathrm{cos}\:{x}−\mathrm{3}\right)\right]\:?\: \\ $$

Question Number 159715 Answers: 0 Comments: 1

Question Number 159693 Answers: 1 Comments: 0

Ω:=∫_1 ^( 10) x d (x + ⌊ x ⌋) =?

$$ \\ $$$$\:\:\:\:\:\:\:\Omega:=\int_{\mathrm{1}} ^{\:\mathrm{10}} {x}\:{d}\:\left({x}\:+\:\lfloor\:{x}\:\rfloor\right)\:=? \\ $$$$ \\ $$

Question Number 159691 Answers: 0 Comments: 1

Question Number 159690 Answers: 1 Comments: 0

Question Number 159683 Answers: 2 Comments: 1

2 ≤ ∣x−2∣ ≤ 6

$$\mathrm{2}\:\leqslant\:\mid\boldsymbol{{x}}−\mathrm{2}\mid\:\leqslant\:\mathrm{6} \\ $$

Question Number 159682 Answers: 3 Comments: 0

∫_( 0) ^( (π/2)) ((cos x sin x)/(cos x + sin x)) dx =?

$$\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{cos}\:{x}\:\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}\:+\:\mathrm{sin}\:{x}}\:{dx}\:=?\: \\ $$

Question Number 159681 Answers: 1 Comments: 0

prove that : P= Π_(n=1) ^∞ (1−(1/(n(n+2))) ) =^? ((−(√2) sin(π(√2) ))/π) m.n

$$ \\ $$$$\:\:\:\:{prove}\:{that}\:: \\ $$$$\mathrm{P}=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}\left({n}+\mathrm{2}\right)}\:\right)\:\overset{?} {=}\:\frac{−\sqrt{\mathrm{2}}\:{sin}\left(\pi\sqrt{\mathrm{2}}\:\right)}{\pi} \\ $$$$\:\:\:\:\:{m}.{n} \\ $$

Question Number 159680 Answers: 0 Comments: 2

∫_( 0) ^( (π/6)) ((sin x sin (x+60°) sin (x+120°))/(cos 3x + sin 3x)) dx=?

$$\:\:\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{6}}} \:\frac{\mathrm{sin}\:{x}\:\mathrm{sin}\:\left({x}+\mathrm{60}°\right)\:\mathrm{sin}\:\left({x}+\mathrm{120}°\right)}{\mathrm{cos}\:\mathrm{3}{x}\:+\:\mathrm{sin}\:\mathrm{3}{x}}\:{dx}=? \\ $$

Question Number 159675 Answers: 1 Comments: 2

Question Number 159671 Answers: 1 Comments: 0

∫ _0 ^∞ ((sin^2 (x)−xsin(x))/x^3 ) dx

$$\int\underset{\mathrm{0}} {\overset{\infty} {\:}}\:\frac{\mathrm{sin}^{\mathrm{2}} \left({x}\right)−{x}\mathrm{sin}\left({x}\right)}{{x}^{\mathrm{3}} }\:{dx} \\ $$

Question Number 159670 Answers: 1 Comments: 0

Question Number 159669 Answers: 1 Comments: 0

find the relative maximum or minimum or neither at the given critical points of the function? f^′ (x)=6x(x^2 −4)^4 (x^2 −1)^2 +8x(x^2 −1)^3 (x^2 −4)^4 , x = 1, x = 2

$${find}\:{the}\:{relative}\:{maximum}\:{or}\:{minimum} \\ $$$${or}\:{neither}\:{at}\:{the}\:{given}\:{critical}\: \\ $$$${points}\:{of}\:{the}\:{function}? \\ $$$${f}^{'} \left({x}\right)=\mathrm{6}{x}\left({x}^{\mathrm{2}} −\mathrm{4}\right)^{\mathrm{4}} \left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} +\mathrm{8}{x}\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{3}} \left({x}^{\mathrm{2}} −\mathrm{4}\right)^{\mathrm{4}} ,\: \\ $$$${x}\:=\:\mathrm{1},\:{x}\:=\:\mathrm{2} \\ $$

Question Number 159668 Answers: 0 Comments: 0

Study the nature of Σ(n^n /((lnn)^n^2 ))

$$\mathrm{Study}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of} \\ $$$$\:\:\:\:\Sigma\frac{{n}^{{n}} }{\left(\mathrm{ln}{n}\right)^{{n}^{\mathrm{2}} } } \\ $$

Question Number 159664 Answers: 0 Comments: 2

prove that : Φ = ∫_0 ^( ∞) (( sin^( 4) (x))/x^( 3) )dx= ln(2) −−−−−−−−−

$$ \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{4}} \left({x}\right)}{{x}^{\:\mathrm{3}} }{dx}=\:\:{ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:−−−−−−−−− \\ $$$$ \\ $$

Question Number 159663 Answers: 0 Comments: 0

find laplace transform for f(t)=(√t) sinh(t) f(t)=(√t) cosh(t)

$${find}\:{laplace}\:{transform}\:{for} \\ $$$${f}\left({t}\right)=\sqrt{{t}}\:{sinh}\left({t}\right) \\ $$$${f}\left({t}\right)=\sqrt{{t}}\:{cosh}\left({t}\right) \\ $$

Question Number 159654 Answers: 0 Comments: 0

Find: 𝛀 =∫_( 0) ^( ∞) ((x ∙ arctan^2 (x))/((x + 1)(x^2 + 1))) dx

$$\mathrm{Find}:\:\:\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{x}\:\centerdot\:\mathrm{arctan}^{\mathrm{2}} \left(\mathrm{x}\right)}{\left(\mathrm{x}\:+\:\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}\right)}\:\mathrm{dx} \\ $$

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