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Question Number 141345 Answers: 0 Comments: 0

((log(ζ(s)))/s)=∫_1 ^∞ J(x)x^(−s−1) dx ( Prove that) Here J(x)=π(x)+(1/2)π((√x))+(1/3)π((x)^(1/3) )+... π(x):=Prime counting function

$$\frac{{log}\left(\zeta\left({s}\right)\right)}{{s}}=\int_{\mathrm{1}} ^{\infty} {J}\left({x}\right){x}^{−{s}−\mathrm{1}} {dx}\:\left(\:{Prove}\:{that}\right) \\ $$$${Here}\:\:{J}\left({x}\right)=\pi\left({x}\right)+\frac{\mathrm{1}}{\mathrm{2}}\pi\left(\sqrt{{x}}\right)+\frac{\mathrm{1}}{\mathrm{3}}\pi\left(\sqrt[{\mathrm{3}}]{{x}}\right)+... \\ $$$$\pi\left({x}\right):=\boldsymbol{\mathrm{P}{rime}}\:\boldsymbol{{counting}}\:\boldsymbol{{function}} \\ $$

Question Number 141340 Answers: 0 Comments: 0

Given the function f defined by f(x) = { ((((2e^x )/(e^x −1)),x≠ 0)),((0, x = 0)) :} (i) study the differentiability of f at x = 0. (ii) Show that the point (0,1) is the centre of symetry to the curve of f.

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function}\:{f}\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:{f}\left({x}\right)\:=\:\begin{cases}{\frac{\mathrm{2}{e}^{{x}} }{{e}^{{x}} −\mathrm{1}},{x}\neq\:\mathrm{0}}\\{\mathrm{0},\:{x}\:=\:\mathrm{0}}\end{cases} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{study}\:\mathrm{the}\:\mathrm{differentiability}\:\mathrm{of}\:{f}\:\mathrm{at}\:{x}\:=\:\mathrm{0}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{0},\mathrm{1}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{symetry}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{curve}\:\mathrm{of}\:{f}. \\ $$

Question Number 141336 Answers: 2 Comments: 0

∫sin (ln (x))dx=? Please

$$\int\mathrm{sin}\:\left(\mathrm{ln}\:\left({x}\right)\right){dx}=?\:\:{Please} \\ $$

Question Number 141417 Answers: 2 Comments: 0

........ advanced ... ... ... calculus....... prove that:: F:= ∫_(−1) ^( 0) ((e^x +e^(1/x) −1)/x) dx=^(??) γ

$$\:\:\:\:\:\:\:\:\:\:........\:{advanced}\:...\:...\:...\:{calculus}....... \\ $$$$\:\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\mathscr{F}:=\:\int_{−\mathrm{1}} ^{\:\mathrm{0}} \frac{{e}^{{x}} +{e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}}{{x}}\:{dx}\overset{??} {=}\gamma \\ $$

Question Number 141414 Answers: 0 Comments: 1

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