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Question Number 53894 Answers: 2 Comments: 0

∫_(π/6) ^(5π/6) (√(4−4 sin^2 t)) dt =

$$\underset{\pi/\mathrm{6}} {\overset{\mathrm{5}\pi/\mathrm{6}} {\int}}\sqrt{\mathrm{4}−\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} {t}}\:{dt}\:= \\ $$

Question Number 53867 Answers: 1 Comments: 0

Question Number 53852 Answers: 0 Comments: 0

G_(μν) = R_(μν) − (1/2) Rg_(μν) + 𝚲g_(μν) Wich theory of modern physic belongs this equation? and what does it mean?

$${G}_{\mu\nu} =\:{R}_{\mu\nu} −\:\frac{\mathrm{1}}{\mathrm{2}}\:{Rg}_{\mu\nu} \:+\:\boldsymbol{\Lambda}{g}_{\mu\nu} \\ $$$$\mathrm{Wich}\:\mathrm{theory}\:\mathrm{of}\:\mathrm{modern}\:\mathrm{physic}\:\mathrm{belongs}\:\mathrm{this}\:\mathrm{equation}? \\ $$$$\mathrm{and}\:\mathrm{what}\:\mathrm{does}\:\mathrm{it}\:\mathrm{mean}? \\ $$

Question Number 53843 Answers: 1 Comments: 0

Question Number 53842 Answers: 0 Comments: 0

Find all the real valued f satisfying f[2x + f(2y)] + f[f(y)] = 4x + 8y for all real numbers x and y.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{real}\:\mathrm{valued}\:\:\mathrm{f}\:\:\mathrm{satisfying}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\left[\mathrm{2x}\:+\:\mathrm{f}\left(\mathrm{2y}\right)\right]\:+\:\mathrm{f}\left[\mathrm{f}\left(\mathrm{y}\right)\right]\:\:=\:\:\mathrm{4x}\:+\:\mathrm{8y} \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{real}\:\mathrm{numbers}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}.\: \\ $$

Question Number 53841 Answers: 0 Comments: 8

Question Number 53839 Answers: 0 Comments: 4

Question Number 53828 Answers: 1 Comments: 1

Question Number 53827 Answers: 1 Comments: 0

5×6^x −3×4^x = 2×9^x I need an explanation.

$$\mathrm{5}×\mathrm{6}^{{x}} −\mathrm{3}×\mathrm{4}^{{x}} =\:\mathrm{2}×\mathrm{9}^{{x}} \\ $$$$ \\ $$$${I}\:{need}\:{an}\:{explanation}. \\ $$

Question Number 53824 Answers: 2 Comments: 1

Question Number 53877 Answers: 2 Comments: 1

2×4^(x+2) −5×4^(x+1) −3×2^(2x+1) −4^x = 20

$$\mathrm{2}×\mathrm{4}^{{x}+\mathrm{2}} −\mathrm{5}×\mathrm{4}^{{x}+\mathrm{1}} −\mathrm{3}×\mathrm{2}^{\mathrm{2}{x}+\mathrm{1}} −\mathrm{4}^{{x}} =\:\mathrm{20} \\ $$

Question Number 53814 Answers: 0 Comments: 0

Question Number 53813 Answers: 0 Comments: 0

Question Number 53811 Answers: 1 Comments: 1

Question Number 53805 Answers: 1 Comments: 4

Question Number 53795 Answers: 1 Comments: 1

calculate Σ_(n=0) ^∞ (((−1)^n )/(4n+1)) =1−(1/5) +(1/9) −(1/(13)) +....

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}+\mathrm{1}}\:=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{5}}\:+\frac{\mathrm{1}}{\mathrm{9}}\:−\frac{\mathrm{1}}{\mathrm{13}}\:+.... \\ $$

Question Number 53792 Answers: 0 Comments: 3

Question Number 53785 Answers: 0 Comments: 1

let f(x)=∫_0 ^∞ ((tsin(tx))/(1+t^4 ))dt with x>0 1) find a explicit form of f(x) 2) find the value of ∫_0 ^∞ ((tsin(2t))/(1+t^4 ))dt.

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{tsin}\left({tx}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{tsin}\left(\mathrm{2}{t}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}. \\ $$

Question Number 53783 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (t^2 /(e^t −1))dt interms of ξ(3)

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{\mathrm{2}} }{{e}^{{t}} −\mathrm{1}}{dt}\:{interms}\:{of}\:\xi\left(\mathrm{3}\right) \\ $$

Question Number 53782 Answers: 1 Comments: 0

calculate ∫_0 ^1 (t^2 /(1+t^3 ))dt

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{3}} }{dt} \\ $$

Question Number 53781 Answers: 1 Comments: 1

calculateA_n =Σ_(n=1) ^∞ x^n cos(nθ) and B_n =Σ_(n=1) ^∞ x^n sin(nθ)

$${calculateA}_{{n}} =\sum_{{n}=\mathrm{1}} ^{\infty} \:{x}^{{n}} {cos}\left({n}\theta\right)\:\:{and}\:{B}_{{n}} =\sum_{{n}=\mathrm{1}} ^{\infty} \:{x}^{{n}} {sin}\left({n}\theta\right) \\ $$

Question Number 53780 Answers: 0 Comments: 0

find nature of the serie Σsin(πen!)

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma{sin}\left(\pi{en}!\right) \\ $$

Question Number 53779 Answers: 0 Comments: 0

Σ u_n is a convergent serie (u_n >0) find nature of the serie 1) Σ ((√u_n )/n) 2)Σ (u_n /(1+u_n ))

$$\Sigma\:{u}_{{n}} {is}\:{a}\:{convergent}\:{serie}\:\left({u}_{{n}} >\mathrm{0}\right)\:\:{find}\:{nature}\:{of}\:{the}\:{serie} \\ $$$$\left.\mathrm{1}\right)\:\Sigma\:\frac{\sqrt{{u}_{{n}} }}{{n}} \\ $$$$\left.\mathrm{2}\right)\Sigma\:\:\frac{{u}_{{n}} }{\mathrm{1}+{u}_{{n}} } \\ $$

Question Number 53778 Answers: 0 Comments: 1

let U_n =(1/(nH_n )) with H_n =Σ_(k=1) ^n (1/k) study the convergence of Σ_(n≥1) U_n 2) study the convergence of Σ_(n≥1) U_n ^2

$${let}\:{U}_{{n}} =\frac{\mathrm{1}}{{nH}_{{n}} }\:\:\:\:{with}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$${study}\:{the}\:{convergence}\:{of}\:\sum_{{n}\geqslant\mathrm{1}} \:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\sum_{{n}\geqslant\mathrm{1}} {U}_{{n}} ^{\mathrm{2}} \\ $$

Question Number 53775 Answers: 0 Comments: 0

let A = (((2 1)),((−1 1)) ) 1) determine P inversible and D diagoanal in ordre to have A =PDP^(−1) 1) calculate A^n with n integr nstural 2) calculate e^(t A) with t ∈ R 3) calculate e^(−A) .

$${let}\:\:{A}\:=\:\begin{pmatrix}{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{−\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:\:{P}\:{inversible}\:{and}\:{D}\:{diagoanal}\:{in}\:{ordre}\:{to}\:{have} \\ $$$${A}\:={PDP}^{−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}^{{n}} \:\:{with}\:{n}\:{integr}\:{nstural} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{e}^{{t}\:{A}} \:\:\:{with}\:{t}\:\in\:{R}\:\: \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{e}^{−{A}} \:\:. \\ $$

Question Number 53770 Answers: 1 Comments: 0

Solve the equation: (he)^2 = she , where h, e and s are integers .

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{he}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{she}\:\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{where}\:\:\mathrm{h},\:\mathrm{e}\:\:\mathrm{and}\:\:\mathrm{s}\:\:\mathrm{are}\:\mathrm{integers}\:. \\ $$

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