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AllQuestion and Answers: Page 716

Question Number 138499 Answers: 1 Comments: 1

Question Number 138495 Answers: 0 Comments: 1

Question Number 138437 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (1/((2n)!!))=(√e)

$$\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2n}\right)!!}=\sqrt{\mathrm{e}} \\ $$

Question Number 138422 Answers: 0 Comments: 0

.......nice calculus..... evaluate: 𝛗=∫_0 ^( ∞) ((((1+x))^(1/3) −(x)^(1/3) )/( (√x)))^( ) dx=?

$$\:\:\:\:\:\:\:\:\:.......{nice}\:\:\:\:\:{calculus}..... \\ $$$$\:\:\:{evaluate}: \\ $$$$\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \:\frac{\sqrt[{\mathrm{3}}]{\mathrm{1}+{x}}\:−\sqrt[{\mathrm{3}}]{{x}}}{\:\sqrt{{x}}}\:^{\:\:} {dx}=? \\ $$$$ \\ $$

Question Number 138417 Answers: 1 Comments: 0

the x axis is transformed with respect to the line y = 2x +2, determine the equation for the image

$$ \\ $$the x axis is transformed with respect to the line y = 2x +2, determine the equation for the image

Question Number 138415 Answers: 1 Comments: 0

A,B ∈R, f(1)=0 , ∫_0 ^1 (f(x))^2 dx =A and ∫_0 ^1 xf(x)dx=B what is the integral value of ∫_0 ^1 xf(x)(f ′(x)−1)dx by using trrms of A and B ?

$${A},{B}\:\in{R},\:\:{f}\left(\mathrm{1}\right)=\mathrm{0}\:,\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\left({f}\left({x}\right)\right)^{\mathrm{2}} {dx}\:={A}\:{and}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{xf}\left({x}\right){dx}={B}\: \\ $$$${what}\:{is}\:{the}\:{integral}\:{value}\:{of}\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{xf}\left({x}\right)\left({f}\:'\left({x}\right)−\mathrm{1}\right){dx}\:{by}\:{using}\:{trrms}\:{of}\:{A}\:{and}\:{B}\:?\: \\ $$

Question Number 138424 Answers: 0 Comments: 1

∣∫_1 ^(√3) ((e^(−x) sin x)/(x^2 +1))dx∣≤(π/(12e))

$$\mid\underset{\mathrm{1}} {\overset{\sqrt{\mathrm{3}}} {\int}}\:\frac{{e}^{−{x}} {sin}\:{x}}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}\mid\leqslant\frac{\pi}{\mathrm{12}{e}} \\ $$

Question Number 138469 Answers: 2 Comments: 0

If x^(26) −521x^(13) =1 x^(10) +(1/x^(10) )=? Any help

$$\boldsymbol{\mathrm{If}}\:\:\:\boldsymbol{\mathrm{x}}^{\mathrm{26}} −\mathrm{521}\boldsymbol{\mathrm{x}}^{\mathrm{13}} =\mathrm{1} \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{x}}^{\mathrm{10}} +\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{10}} }=? \\ $$$$\boldsymbol{\mathrm{Any}}\:\boldsymbol{\mathrm{help}}\: \\ $$

Question Number 138428 Answers: 2 Comments: 0

what the area of area bounded by line y= ∣ln x∣ and y= 2

$${what}\:{the}\:{area}\:{of}\:\:{area}\:{bounded}\:{by}\:{line} \\ $$$${y}=\:\mid{ln}\:{x}\mid\:{and}\:{y}=\:\mathrm{2}\: \\ $$

Question Number 138407 Answers: 1 Comments: 0

if the F(x)=(1/x)∫_1 ^x (2t−F ′(t))dt ⇒ what the F ′(1) value using the Leibnitz formula.

$${if}\:{the}\:{F}\left({x}\right)=\frac{\mathrm{1}}{{x}}\underset{\mathrm{1}} {\overset{{x}} {\int}}\left(\mathrm{2}{t}−{F}\:'\left({t}\right)\right){dt}\:\:\Rightarrow\:{what}\:{the}\:{F}\:'\left(\mathrm{1}\right)\:{value}\:{using}\:{the}\:{Leibnitz}\:{formula}. \\ $$

Question Number 138403 Answers: 1 Comments: 0

∫(e^(4t) /(e^(2t) +3e^t +2))dt=?

$$\int\frac{{e}^{\mathrm{4}{t}} }{{e}^{\mathrm{2}{t}} +\mathrm{3}{e}^{{t}} +\mathrm{2}}{dt}=? \\ $$

Question Number 138401 Answers: 0 Comments: 5

Question Number 138391 Answers: 1 Comments: 0

x^3 −x−c=0 ; 0<c<(2/(3(√3))) Find x.

$${x}^{\mathrm{3}} −{x}−{c}=\mathrm{0}\:\:;\:\mathrm{0}<{c}<\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{3}}} \\ $$$${Find}\:{x}. \\ $$

Question Number 138388 Answers: 2 Comments: 0

Find maximum volume of a cylinder within a unit cube whose axis is along a diagonal of the cube.

$${Find}\:{maximum}\:{volume}\:{of}\:{a} \\ $$$${cylinder}\:{within}\:{a}\:{unit}\:{cube}\: \\ $$$${whose}\:{axis}\:{is}\:{along}\:{a}\:{diagonal} \\ $$$${of}\:{the}\:{cube}. \\ $$

Question Number 138386 Answers: 1 Comments: 1

Question Number 138383 Answers: 0 Comments: 2

hi ! for a_0 = 1 and ∀ n ≥ 1, a_n = (1/n) Σ_(k=0) ^(n−1) (a_k /(n−k)) . prove that ∀ n ≥ 0, we get 0 ≤ a_n ≤ 1.

$$\boldsymbol{\mathrm{hi}}\:! \\ $$$$\boldsymbol{\mathrm{for}}\:{a}_{\mathrm{0}} \:=\:\mathrm{1}\:\boldsymbol{\mathrm{and}}\:\forall\:{n}\:\geqslant\:\mathrm{1},\:{a}_{{n}} \:=\:\frac{\mathrm{1}}{{n}}\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\:\:\frac{{a}_{{k}} }{{n}−{k}}\:. \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\forall\:{n}\:\geqslant\:\mathrm{0},\:\boldsymbol{\mathrm{we}}\:\boldsymbol{\mathrm{get}}\:\mathrm{0}\:\leqslant\:{a}_{{n}} \:\leqslant\:\mathrm{1}. \\ $$

Question Number 138379 Answers: 0 Comments: 5

Question Number 138378 Answers: 1 Comments: 0

I(t,s)=∫_0 ^∞ x^(−t) (1+x)^(−s) dx=((Γ(1−t)Γ(s+t−1))/(Γ(s))) I(t,s)=∫_0 ^∞ x^(−t) (1+x)^(−s) dx =∫_0 ^∞ x^(−t) (1+x)^(m−s) (1+x)^(−m) dx =∫_0 ^∞ x^(−t) e^((m−s)ln(1+x)) (1+x)^(−m) dx =Σ_(n=0) ^∞ (((m−s)^n )/(n!))∫_0 ^∞ ((ln^n (1+x))/(x^t (1+x)^m ))dx =2Σ_(n=0) ^∞ (((m−s)^n )/(n!))∫_0 ^∞ ((ln^n (1+x^2 ))/((1+x^2 )^m ))dx (t=(1/2)) If i want to calculate ∫_0 ^∞ ((ln^n (1+x^2 ))/((1+x^2 )^m )) how to expand “ ((Γ(1−t)Γ(s+t−1))/(Γ(s))) ”into summation?

$${I}\left({t},{s}\right)=\int_{\mathrm{0}} ^{\infty} {x}^{−{t}} \left(\mathrm{1}+{x}\right)^{−{s}} {dx}=\frac{\Gamma\left(\mathrm{1}−{t}\right)\Gamma\left({s}+{t}−\mathrm{1}\right)}{\Gamma\left({s}\right)} \\ $$$${I}\left({t},{s}\right)=\int_{\mathrm{0}} ^{\infty} {x}^{−{t}} \left(\mathrm{1}+{x}\right)^{−{s}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\int_{\mathrm{0}} ^{\infty} {x}^{−{t}} \left(\mathrm{1}+{x}\right)^{{m}−{s}} \left(\mathrm{1}+{x}\right)^{−{m}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\int_{\mathrm{0}} ^{\infty} {x}^{−{t}} {e}^{\left({m}−{s}\right){ln}\left(\mathrm{1}+{x}\right)} \left(\mathrm{1}+{x}\right)^{−{m}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left({m}−{s}\right)^{{n}} }{{n}!}\int_{\mathrm{0}} ^{\infty} \frac{{ln}^{{n}} \left(\mathrm{1}+{x}\right)}{{x}^{{t}} \left(\mathrm{1}+{x}\right)^{{m}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left({m}−{s}\right)^{{n}} }{{n}!}\int_{\mathrm{0}} ^{\infty} \frac{{ln}^{{n}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{m}} }{dx}\:\:\:\:\:\:\:\:\:\:\:\:\left({t}=\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$${If}\:{i}\:{want}\:{to}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \frac{{ln}^{{n}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{m}} } \\ $$$${how}\:{to}\:{expand}\:\:``\:\frac{\Gamma\left(\mathrm{1}−{t}\right)\Gamma\left({s}+{t}−\mathrm{1}\right)}{\Gamma\left({s}\right)}\:''{into}\:{summation}? \\ $$

Question Number 138377 Answers: 0 Comments: 0

...... advanced ... ... ... calculus...... evaluate:::: 𝛗=∫_0 ^( ∞) ((ln(x))/( (x^2 )^(1/3) .(2+x))) dx=??

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:......\:{advanced}\:...\:...\:...\:{calculus}...... \\ $$$$\:\:\:{evaluate}:::: \\ $$$$\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left({x}\right)}{\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} }\:.\left(\mathrm{2}+{x}\right)}\:{dx}=?? \\ $$$$\:\: \\ $$

Question Number 138376 Answers: 0 Comments: 0

log_4 (x+2)+log_3 (x−2)=1 find x

$$\boldsymbol{\mathrm{log}}_{\mathrm{4}} \left(\boldsymbol{\mathrm{x}}+\mathrm{2}\right)+\boldsymbol{\mathrm{log}}_{\mathrm{3}} \left(\boldsymbol{\mathrm{x}}−\mathrm{2}\right)=\mathrm{1} \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{x}} \\ $$

Question Number 138396 Answers: 2 Comments: 2