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Question Number 150721 Answers: 2 Comments: 0

calculate the convergence interval of the serie Σ_(n=0) ^∞ (((−1)^n x^(2n) )/(n!))

$${calculate}\:{the}\:{convergence}\:{interval}\:{of}\:{the} \\ $$$${serie} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{{n}!} \\ $$

Question Number 150226 Answers: 2 Comments: 0

prove that :: ζ (0 )=^? ((−1)/2) ..........■ m.n...

$$ \\ $$$$\:\:\:\:\:\mathrm{prove}\:\:\mathrm{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\zeta\:\left(\mathrm{0}\:\right)\overset{?} {=}\:\frac{−\mathrm{1}}{\mathrm{2}}\:..........\blacksquare \\ $$$$\:\:\:\:\:\:\:{m}.{n}... \\ $$

Question Number 150121 Answers: 1 Comments: 0

Find in closed form: n∈N^∗ ∫_( 0) ^( 1) ln(1 - x^2 )ln^n (1 - x) dx = ?

$$\mathrm{Find}\:\mathrm{in}\:\mathrm{closed}\:\mathrm{form}:\:\:\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{ln}\left(\mathrm{1}\:-\:\mathrm{x}^{\mathrm{2}} \right)\mathrm{ln}^{\boldsymbol{\mathrm{n}}} \left(\mathrm{1}\:-\:\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 150119 Answers: 2 Comments: 0

Find the smallest value of a given expression: (x^2 + 6x + 8)^2 + 5

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}\:\mathrm{given} \\ $$$$\mathrm{expression}: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{6x}\:+\:\mathrm{8}\right)^{\mathrm{2}} \:+\:\mathrm{5} \\ $$

Question Number 150103 Answers: 1 Comments: 0

Ω = ∫_0 ^( π) sin^( (1/2)) (x). ln( sin (x) )dx=?

$$\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\pi} {sin}^{\:\frac{\mathrm{1}}{\mathrm{2}}} \left({x}\right).\:\mathrm{ln}\left(\:{sin}\:\left({x}\right)\:\right){dx}=? \\ $$$$ \\ $$

Question Number 150097 Answers: 1 Comments: 0

Question Number 150079 Answers: 2 Comments: 0

∫(dx/((x^2 +x+1)^2 ))

$$\int\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 150064 Answers: 1 Comments: 0

How can i evaluate the value of ∫_2 ^( 4) (e^t /t)dt = ?

$$\mathrm{How}\:\mathrm{can}\:\mathrm{i}\:\mathrm{evaluate}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\: \\ $$$$\:\:\int_{\mathrm{2}} ^{\:\mathrm{4}} \frac{\mathrm{e}^{\mathrm{t}} }{\mathrm{t}}\mathrm{dt}\:=\:? \\ $$

Question Number 150063 Answers: 0 Comments: 0

Question Number 150062 Answers: 2 Comments: 0

Question Number 150059 Answers: 2 Comments: 0

Σ_(n=1) ^∞ (1/(n∙(2n + 1))) = ?

$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}\centerdot\left(\mathrm{2n}\:+\:\mathrm{1}\right)}\:=\:? \\ $$

Question Number 150058 Answers: 2 Comments: 2

Solve the equation: cos^4 (x) + i sin^4 (x) = 4e^(4ix)

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{cos}^{\mathrm{4}} \left(\mathrm{x}\right)\:+\:\mathrm{i}\:\mathrm{sin}^{\mathrm{4}} \left(\mathrm{x}\right)\:=\:\mathrm{4e}^{\mathrm{4}\boldsymbol{\mathrm{ix}}} \\ $$

Question Number 150056 Answers: 2 Comments: 1

Question Number 150053 Answers: 0 Comments: 0

let x;y;z;t>0 and x+y+z+t=4 prove that (4/((xyzt)^2 )) + 3 ≥ (√(45 + x^4 + y^4 + z^4 + t^4 ))

$$\mathrm{let}\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{t}=\mathrm{4} \\ $$$$\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{4}}{\left(\mathrm{xyzt}\right)^{\mathrm{2}} }\:+\:\mathrm{3}\:\geqslant\:\sqrt{\mathrm{45}\:+\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{z}^{\mathrm{4}} \:+\:\mathrm{t}^{\mathrm{4}} } \\ $$

Question Number 150045 Answers: 0 Comments: 0

Question Number 150044 Answers: 2 Comments: 0

Given f(((2x−3)/(2x+1)))+f(((2x+3)/(1−2x)))= 4x f(x)=?

$$\:\mathrm{Given}\:\mathrm{f}\left(\frac{\mathrm{2x}−\mathrm{3}}{\mathrm{2x}+\mathrm{1}}\right)+\mathrm{f}\left(\frac{\mathrm{2x}+\mathrm{3}}{\mathrm{1}−\mathrm{2x}}\right)=\:\mathrm{4x} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 150040 Answers: 0 Comments: 0

Let a,b,c be positive real numbers such that a+b+c=1 .Prove that ((ab)/(1−c^2 )) +((bc)/(1−a^2 ))+((ca)/(1−b^2 )) ≤ (3/8)

$$\mathrm{Let}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{1}\:.\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\:\frac{\mathrm{ab}}{\mathrm{1}−\mathrm{c}^{\mathrm{2}} }\:+\frac{\mathrm{bc}}{\mathrm{1}−\mathrm{a}^{\mathrm{2}} }+\frac{\mathrm{ca}}{\mathrm{1}−\mathrm{b}^{\mathrm{2}} }\:\leqslant\:\frac{\mathrm{3}}{\mathrm{8}} \\ $$

Question Number 150039 Answers: 1 Comments: 0

Prove that (a/b)+(b/c)+(c/a)≥(√((a^2 +1)/(b^2 +1)))+(√((b^2 +1)/(c^2 +1)))+(√((c^2 +1)/(a^2 +1))) for a,b,c are positive real number

$$\mathrm{Prove}\:\mathrm{that}\:\frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{c}}+\frac{\mathrm{c}}{\mathrm{a}}\geqslant\sqrt{\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{1}}{\mathrm{b}^{\mathrm{2}} +\mathrm{1}}}+\sqrt{\frac{\mathrm{b}^{\mathrm{2}} +\mathrm{1}}{\mathrm{c}^{\mathrm{2}} +\mathrm{1}}}+\sqrt{\frac{\mathrm{c}^{\mathrm{2}} +\mathrm{1}}{\mathrm{a}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}\: \\ $$

Question Number 150037 Answers: 0 Comments: 0

Random Problem: ∫_(π/4) ^(π/2) (−7sin x + 3cos x) dx By getting the antiderivative of the trigonometric functions: ∫ sin(x) dx = −cos x + c ∫ cos(x) dx = sin x + c = −7 ∫ sin x + 3 ∫ cos x ∣_(π/4) ^(π/2) = −7(− cos x) + 3(sin x) ∣_(π/4) ^(π/2) = 7 cos x + 3sin x ∣_(π/4) ^(π/2) Evaluate it to the top and bottom limit of integration: = (7 cos ∙ (π/2) + 3 sin ∙ (π/2))− (7 cos ∙ (π/(4 )) + 3 sin ∙ (π/4) ) =[7(0) + 3(1)] − [7(((√2)/2)) + 3(((√2)/2))] = 3 − ((7(√2))/2) − ((3(√2))/2) = 3 − ((10(√2))/2) or 3 − 5(√2) Answer: 3 − 5(√2) Solution by Roswel:)

$${Random}\:{Problem}: \\ $$$$\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\left(−\mathrm{7sin}\:{x}\:+\:\mathrm{3cos}\:{x}\right)\:{dx} \\ $$$$ \\ $$$${By}\:{getting}\:{the}\:{antiderivative}\:{of}\:{the}\:{trigonometric}\:{functions}: \\ $$$$\int\:\mathrm{sin}\left({x}\right)\:{dx}\:=\:−\mathrm{cos}\:{x}\:+\:{c} \\ $$$$\int\:\mathrm{cos}\left({x}\right)\:{dx}\:=\:\mathrm{sin}\:{x}\:+\:{c} \\ $$$$=\:−\mathrm{7}\:\int\:\mathrm{sin}\:{x}\:\:+\:\:\mathrm{3}\:\int\:\mathrm{cos}\:{x}\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}}\:=\:−\mathrm{7}\left(−\:\mathrm{cos}\:{x}\right)\:+\:\mathrm{3}\left(\mathrm{sin}\:{x}\right)\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}} \\ $$$$=\:\mathrm{7}\:\mathrm{cos}\:{x}\:+\:\mathrm{3sin}\:{x}\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}} \\ $$$$ \\ $$$${Evaluate}\:{it}\:{to}\:{the}\:{top}\:{and}\:{bottom}\:{limit}\:{of}\:{integration}: \\ $$$$ \\ $$$$=\:\left(\mathrm{7}\:\mathrm{cos}\:\centerdot\:\frac{\pi}{\mathrm{2}}\:+\:\mathrm{3}\:\mathrm{sin}\:\centerdot\:\frac{\pi}{\mathrm{2}}\right)−\:\left(\mathrm{7}\:\mathrm{cos}\:\centerdot\:\frac{\pi}{\mathrm{4}\:}\:\:+\:\mathrm{3}\:\mathrm{sin}\:\centerdot\:\frac{\pi}{\mathrm{4}}\:\right) \\ $$$$=\left[\mathrm{7}\left(\mathrm{0}\right)\:+\:\mathrm{3}\left(\mathrm{1}\right)\right]\:−\:\left[\mathrm{7}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\:+\:\mathrm{3}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\right] \\ $$$$=\:\mathrm{3}\:−\:\frac{\mathrm{7}\sqrt{\mathrm{2}}}{\mathrm{2}}\:−\:\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$=\:\mathrm{3}\:−\:\frac{\mathrm{10}\sqrt{\mathrm{2}}}{\mathrm{2}}\:{or}\:\mathrm{3}\:−\:\mathrm{5}\sqrt{\mathrm{2}} \\ $$$$ \\ $$$${Answer}:\:\mathrm{3}\:−\:\mathrm{5}\sqrt{\mathrm{2}} \\ $$$$ \\ $$$$\left.{Solution}\:{by}\:{Roswel}:\right) \\ $$

Question Number 150034 Answers: 1 Comments: 0

Find a closed form: a∈R and a≠0 Ω(a)=∫_( 0) ^( ∞) (x^4 /((1+x^2 )(1+a^4 x^4 ))) dx

$$\mathrm{Find}\:\boldsymbol{\mathrm{a}}\:\mathrm{closed}\:\mathrm{form}:\:\:\boldsymbol{\mathrm{a}}\in\mathbb{R}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{a}}\neq\mathrm{0} \\ $$$$\Omega\left(\mathrm{a}\right)=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\frac{\mathrm{x}^{\mathrm{4}} }{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)\left(\mathrm{1}+\mathrm{a}^{\mathrm{4}} \mathrm{x}^{\mathrm{4}} \right)}\:\mathrm{dx} \\ $$

Question Number 150033 Answers: 1 Comments: 0

Question Number 150030 Answers: 0 Comments: 0

Roll a fair die twice and define A to be event that the sum of the scores showing up is greater than 7, B be the event that the sum of the scores showing up is a multiple of 3 and C be the event that the sum of the scores showing up is a prime number. Which of the events A,B and C are independent event? are the 3 events jointly independent?

Question Number 150029 Answers: 1 Comments: 0

Calcular: (√(8 + 2(√(8 + 2(√(8 + ...)))))) = ?

$$\mathrm{Calcular}: \\ $$$$\sqrt{\mathrm{8}\:+\:\mathrm{2}\sqrt{\mathrm{8}\:+\:\mathrm{2}\sqrt{\mathrm{8}\:+\:...}}}\:=\:? \\ $$

Question Number 150017 Answers: 0 Comments: 2

Question Number 150009 Answers: 1 Comments: 0

Question Number 150007 Answers: 1 Comments: 0

∫ ((x^2 + x)/(x^6 + 1)) dx = ?

$$\int\:\:\frac{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}}{\mathrm{x}^{\mathrm{6}} \:+\:\mathrm{1}}\:\mathrm{dx}\:=\:? \\ $$

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