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Question Number 141413 Answers: 1 Comments: 0

∫^( +∞) _( 1) ((1/(E(x)))−(1/x))dx=???

$$\underset{\:\mathrm{1}} {\int}^{\:+\infty} \left(\frac{\mathrm{1}}{\mathrm{E}\left(\mathrm{x}\right)}−\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{dx}=??? \\ $$

Question Number 141333 Answers: 1 Comments: 0

∫x^2 cos ((x/2))dx

$$\int{x}^{\mathrm{2}} \mathrm{cos}\:\left(\frac{{x}}{\mathrm{2}}\right){dx} \\ $$

Question Number 141323 Answers: 1 Comments: 0

Question Number 141322 Answers: 0 Comments: 0

......advanced........calculus....... prove that:: ξ:=Π_(n=2) ^∞ e(1−(1/n^2 ))^n^2 =(π/(e(√e)))

$$\:\:\:\:\:......{advanced}........{calculus}....... \\ $$$$\:{prove}\:{that}:: \\ $$$$\:\:\:\xi:=\underset{{n}=\mathrm{2}} {\overset{\infty} {\prod}}{e}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)^{{n}^{\mathrm{2}} } =\frac{\pi}{{e}\sqrt{{e}}} \\ $$$$ \\ $$

Question Number 141320 Answers: 1 Comments: 0

......nice ......calculuus..... prove that:: 𝛗:=∫_0 ^( ∞) ∫_0 ^( ∞) ((A rctan(x^2 y^2 ))/(x^4 +y^4 ))dxdy=((π^2 (√2))/(16)) .....

$$\:\:\:\:\:\:\:......{nice}\:......{calculuus}..... \\ $$$$\:\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \int_{\mathrm{0}} ^{\:\infty} \frac{\mathscr{A}\:{rctan}\left({x}^{\mathrm{2}} {y}^{\mathrm{2}} \right)}{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} }{dxdy}=\frac{\pi^{\mathrm{2}} \sqrt{\mathrm{2}}}{\mathrm{16}} \\ $$$$..... \\ $$

Question Number 141319 Answers: 2 Comments: 0

prove:: Ω:=∫_0 ^( 1) ((ln^2 (x))/(1−x^4 ))dx =(π^3 /(32))+(7/8)ζ(3)..

$$\:\:\: \\ $$$$\:\:\:\:\:{prove}:: \\ $$$$\:\:\:\:\:\:\:\Omega:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} \left({x}\right)}{\mathrm{1}−{x}^{\mathrm{4}} }{dx}\:=\frac{\pi^{\mathrm{3}} }{\mathrm{32}}+\frac{\mathrm{7}}{\mathrm{8}}\zeta\left(\mathrm{3}\right).. \\ $$

Question Number 141313 Answers: 0 Comments: 0

Question Number 141383 Answers: 1 Comments: 0

sin^2 (((f(x))/2))=1−(1/2)(√(1−x^2 )) find f(x)

$$\mathrm{sin}^{\mathrm{2}} \left(\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{2}}\right)=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} } \\ $$$$\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$

Question Number 141373 Answers: 1 Comments: 1

Question Number 141372 Answers: 3 Comments: 0

Question Number 141368 Answers: 2 Comments: 0

A closed cylindrical can be is to hold 1 liters of liquid . How should we choose the height and radius to minimize the amount of material needed to manufacture the can ?

$${A}\:{closed}\:{cylindrical}\:{can}\:{be}\:{is}\:{to}\:{hold} \\ $$$$\mathrm{1}\:{liters}\:{of}\:{liquid}\:.\:{How}\:{should}\:{we}\: \\ $$$${choose}\:{the}\:{height}\:{and}\:{radius}\: \\ $$$${to}\:{minimize}\:{the}\:{amount}\:{of} \\ $$$${material}\:{needed}\:{to}\:{manufacture} \\ $$$${the}\:{can}\:?\: \\ $$

Question Number 141367 Answers: 1 Comments: 0

∫((√(cosx∙senx)))dx

$$\int\left(\sqrt{{cosx}\centerdot{senx}}\right){dx} \\ $$

Question Number 141311 Answers: 0 Comments: 0

Show that ,C_n ^k +C_n ^(k−1) =C_(n+1) ^(n−k)

$$\mathrm{Show}\:\mathrm{that}\:,\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} +\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}−\mathrm{1}} =\mathrm{C}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{n}−\mathrm{k}} \\ $$

Question Number 141308 Answers: 1 Comments: 0

Question Number 141303 Answers: 3 Comments: 1

Question Number 141304 Answers: 0 Comments: 1

Question Number 141388 Answers: 1 Comments: 0

∫_0 ^(π/2) (√((senx∙cosx)))dx Help

$$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \sqrt{\left({senx}\centerdot{cosx}\right)}{dx} \\ $$$${Help} \\ $$

Question Number 141387 Answers: 1 Comments: 0

∫_(−π/4) ^(π/4) (sec^2 x+tgx)^2 dx

$$\int_{−\pi/\mathrm{4}} ^{\pi/\mathrm{4}} \left({sec}^{\mathrm{2}} {x}+{tgx}\right)^{\mathrm{2}} {dx} \\ $$

Question Number 141380 Answers: 0 Comments: 0

Question Number 141378 Answers: 2 Comments: 0

prove that:: Π_(n=0) ^∞ (((5n+2)(5n+3))/((5n+1)(5n+4))) =ϕ ϕ:= ((1+(√5))/2)

$$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\frac{\left(\mathrm{5}{n}+\mathrm{2}\right)\left(\mathrm{5}{n}+\mathrm{3}\right)}{\left(\mathrm{5}{n}+\mathrm{1}\right)\left(\mathrm{5}{n}+\mathrm{4}\right)}\:=\varphi\: \\ $$$$\:\:\:\:\:\:\:\varphi:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$

Question Number 141294 Answers: 5 Comments: 0

Find max & min value of f(x)=(x/(x^2 −5x+9)).

$${Find}\:{max}\:\&\:{min}\:{value}\:{of} \\ $$$$\:{f}\left({x}\right)=\frac{{x}}{{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{9}}. \\ $$

Question Number 141328 Answers: 2 Comments: 0

...... Evaluate: F :=Σ_(n=2) ^∞ (((−1)^n ζ(n))/(n+1)) =? .......

$$......\:{Evaluate}: \\ $$$$\:\:\:\:\:\mathscr{F}\::=\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{{n}+\mathrm{1}}\:=? \\ $$$$....... \\ $$

Question Number 141312 Answers: 0 Comments: 0

Σ_(n=1) ^∞ (−1)^n ((Si(2πn)−(π/2))/n)=?

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{{Si}\left(\mathrm{2}\pi{n}\right)−\frac{\pi}{\mathrm{2}}}{{n}}=? \\ $$

Question Number 141289 Answers: 1 Comments: 1

Question Number 141381 Answers: 0 Comments: 0

Let a,b ≥ 0 . Prove that (a+b+2)^3 ≥ ((27)/2)(a^2 +ab+b^2 )

$$\mathrm{Let}\:\:{a},{b}\:\geqslant\:\mathrm{0}\:.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}+{b}+\mathrm{2}\right)^{\mathrm{3}} \:\geqslant\:\frac{\mathrm{27}}{\mathrm{2}}\left({a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} \right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 141291 Answers: 0 Comments: 3

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