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AllQuestion and Answers: Page 35

Question Number 219945 Answers: 2 Comments: 0

Question Number 219944 Answers: 7 Comments: 0

Question Number 219940 Answers: 1 Comments: 0

Let: f : [n−1 , n] → [n , n + 1] be a continuous function Such that: ∫_(n−1) ^( n) (1 + xf^′ (x))dx ≤ nf(n)−(n−1)f(n−1) Then prove: ∫_(n−1) ^( n) (dx/(f(x))) ≤ (2/(n + 1)) , n∈N^∗

$$\mathrm{Let}: \\ $$$$\mathrm{f}\::\:\left[\mathrm{n}−\mathrm{1}\:,\:\mathrm{n}\right]\:\rightarrow\:\left[\mathrm{n}\:,\:\mathrm{n}\:+\:\mathrm{1}\right] \\ $$$$\mathrm{be}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function} \\ $$$$\mathrm{Such}\:\mathrm{that}: \\ $$$$\int_{\boldsymbol{\mathrm{n}}−\mathrm{1}} ^{\:\boldsymbol{\mathrm{n}}} \left(\mathrm{1}\:+\:\mathrm{xf}\:^{'} \left(\mathrm{x}\right)\right)\mathrm{dx}\:\leqslant\:\mathrm{nf}\left(\mathrm{n}\right)−\left(\mathrm{n}−\mathrm{1}\right)\mathrm{f}\left(\mathrm{n}−\mathrm{1}\right) \\ $$$$\mathrm{Then}\:\mathrm{prove}: \\ $$$$\int_{\boldsymbol{\mathrm{n}}−\mathrm{1}} ^{\:\boldsymbol{\mathrm{n}}} \:\frac{\mathrm{dx}}{\mathrm{f}\left(\mathrm{x}\right)}\:\leqslant\:\frac{\mathrm{2}}{\mathrm{n}\:+\:\mathrm{1}}\:\:\:,\:\:\:\mathrm{n}\in\mathbb{N}^{\ast} \\ $$

Question Number 219936 Answers: 1 Comments: 0

Prove that:∀n∈IN ∫^( n+1) _( n) ln(t)dt≤ln(n+(1/2))

$$\mathrm{Prove}\:\mathrm{that}:\forall\mathrm{n}\in\mathrm{IN} \\ $$$$\underset{\:\mathrm{n}} {\int}^{\:\mathrm{n}+\mathrm{1}} \mathrm{ln}\left(\mathrm{t}\right)\mathrm{dt}\leqslant\mathrm{ln}\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 219935 Answers: 1 Comments: 0

Question Number 219933 Answers: 1 Comments: 0

A^1 + B^2 + C^3 + D^4 = ABCD^(−) find ABCD

$$\mathrm{A}^{\mathrm{1}} \:+\:\mathrm{B}^{\mathrm{2}} \:+\:\mathrm{C}^{\mathrm{3}} \:+\:\mathrm{D}^{\mathrm{4}} \:=\:\overline {\mathrm{ABCD}} \\ $$$${find}\:\:{ABCD} \\ $$

Question Number 220100 Answers: 1 Comments: 0

Question Number 219921 Answers: 5 Comments: 0

Question Number 219919 Answers: 4 Comments: 0

Question Number 219918 Answers: 6 Comments: 2

Question Number 219917 Answers: 0 Comments: 0

Question Number 219911 Answers: 1 Comments: 1

∫_( 0) ^( ∞) (Σ_(n≥1) ((sin(2πnx))/n))(dx/x^(s+1) )

$$ \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\infty} \left(\underset{{n}\geqslant\mathrm{1}} {\sum}\:\frac{{sin}\left(\mathrm{2}\pi{nx}\right)}{{n}}\right)\frac{{dx}}{{x}^{{s}+\mathrm{1}} } \\ $$$$ \\ $$

Question Number 219899 Answers: 3 Comments: 0

Question Number 219898 Answers: 1 Comments: 0

Question Number 219890 Answers: 1 Comments: 0

Find Maxima x+y where x^2 +y^2 =r^2 (use Lagrange Method)

$$\mathrm{Find}\:\mathrm{Maxima}\: \\ $$$${x}+{y}\:\mathrm{where}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={r}^{\mathrm{2}} \:\left(\mathrm{use}\:\mathrm{Lagrange}\:\mathrm{Method}\right) \\ $$

Question Number 219887 Answers: 1 Comments: 0

what is lim_(n→∞) (1+(1/n))↑↑^∞ =?? a↑↑^m =a^a^a^a^⋰ _(m times) (aka Knuth′s up notation)

$$\mathrm{what}\:\mathrm{is}\: \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)\uparrow\uparrow^{\infty} =?? \\ $$$${a}\uparrow\uparrow^{{m}} =\underset{{m}\:\mathrm{times}} {\underbrace{{a}^{{a}^{{a}^{{a}^{\iddots} } } } }}\:\:\left(\mathrm{aka}\:\mathrm{Knuth}'\mathrm{s}\:\mathrm{up}\:\mathrm{notation}\right) \\ $$

Question Number 219884 Answers: 2 Comments: 0

(a,b,c)>0 such that, a+b+c=13, abc=36 find the maximum and minimum value of ab+bc+ca=?

$$\:\left({a},{b},{c}\right)>\mathrm{0}\:{such}\:{that}, \\ $$$$\:\:{a}+{b}+{c}=\mathrm{13},\:\:{abc}=\mathrm{36} \\ $$$$\:\:{find}\:{the}\:{maximum}\:{and}\:{minimum}\: \\ $$$$\:{value}\:{of}\:\:{ab}+{bc}+{ca}=? \\ $$

Question Number 219879 Answers: 1 Comments: 0

Question Number 219874 Answers: 2 Comments: 0

find (√2^6^2^1^4^4 )=?

$${find}\:\sqrt{\mathrm{2}^{\mathrm{6}^{\mathrm{2}^{\mathrm{1}^{\mathrm{4}^{\mathrm{4}} } } } } }=? \\ $$

Question Number 219872 Answers: 1 Comments: 0

prove ∫ Y_(−(3/2)) (z) dz=((4sin(z)+((z𝚪((1/2),−iz))/( (√(−iz))))+((z𝚪((1/2),iz))/( (√(iz)))))/( (√(2πz))))+C

$$\mathrm{prove} \\ $$$$\int\:\:{Y}_{−\frac{\mathrm{3}}{\mathrm{2}}} \left({z}\right)\:\mathrm{d}{z}=\frac{\mathrm{4sin}\left({z}\right)+\frac{{z}\boldsymbol{\Gamma}\left(\frac{\mathrm{1}}{\mathrm{2}},−\boldsymbol{{i}}{z}\right)}{\:\sqrt{−\boldsymbol{{i}}{z}}}+\frac{{z}\boldsymbol{\Gamma}\left(\frac{\mathrm{1}}{\mathrm{2}},\boldsymbol{{i}}{z}\right)}{\:\sqrt{\boldsymbol{{i}}{z}}}}{\:\sqrt{\mathrm{2}\pi{z}}}+{C} \\ $$

Question Number 219870 Answers: 1 Comments: 0

∫_0 ^( ∞) K_ν (r)dr ∫_0 ^( ∞) t∙Y_0 (t)dt ∫_0 ^( ∞) ((sin(t)e^(−kt) )/(t^2 +ρ^2 ))dt

$$\int_{\mathrm{0}} ^{\:\infty} \:{K}_{\nu} \left({r}\right)\mathrm{d}{r} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:{t}\centerdot{Y}_{\mathrm{0}} \left({t}\right)\mathrm{d}{t} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{\mathrm{sin}\left({t}\right){e}^{−{kt}} }{{t}^{\mathrm{2}} +\rho^{\mathrm{2}} }\mathrm{d}{t}\: \\ $$

Question Number 219869 Answers: 1 Comments: 0

Question Number 219868 Answers: 1 Comments: 0

Prove that; (d/dx) (((sin^( 2) x)/(1+cot x)) + ((cos^( 2) x)/(1+tan x))) = −cos 2x

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\frac{{d}}{{dx}}\:\left(\frac{\mathrm{sin}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{cot}\:{x}}\:+\:\frac{\mathrm{cos}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{tan}\:{x}}\right)\:=\:−\mathrm{cos}\:\mathrm{2}{x}\:\:\:\: \\ $$$$ \\ $$

Question Number 219866 Answers: 1 Comments: 0

Prove that; ∫^( π/2) _( 0) sin^(2x−1) θ cos^(2y−1) θ dθ = (1/2) ((Γ(x)Γ(y))/(Γ(x)+Γ(y)))

$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\pi/\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}{x}−\mathrm{1}} \theta\:\mathrm{cos}\:^{\mathrm{2}{y}−\mathrm{1}} \theta\:{d}\theta\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}\right)+\Gamma\left({y}\right)}\:\:\:\:\: \\ $$$$\: \\ $$

Question Number 219865 Answers: 1 Comments: 0

Prove that; ∫_( 0) ^( 1) lnΓ(x)dx = ln (√(2π))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:{ln}\Gamma\left({x}\right){dx}\:=\:{ln}\:\sqrt{\mathrm{2}\pi} \\ $$$$ \\ $$

Question Number 219864 Answers: 1 Comments: 0

Prove that; ∫_( 0) ^( 1) (x^( n+1) /(x+1)) dx < (1/(2(n+1)))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}^{\:{n}+\mathrm{1}} }{{x}+\mathrm{1}}\:{dx}\:<\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$$$ \\ $$

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