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Question Number 158354    Answers: 0   Comments: 2

(√(sin ((π/4)+x))) +(√(sin ((π/4)−x))) =((2cos 2x))^(1/4)

$$\:\sqrt{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+{x}\right)}\:+\sqrt{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}−{x}\right)}\:=\sqrt[{\mathrm{4}}]{\mathrm{2cos}\:\mathrm{2}{x}} \\ $$

Question Number 158341    Answers: 1   Comments: 0

Question Number 158340    Answers: 2   Comments: 0

1) Proven that by all n ∈ N^∗ 2!4!..(2n)!≥((n+1)!)^n 2) Proven by recurring that Σ_(p=1) ^n pp!=(n+1)!−1

$$\left.\mathrm{1}\right)\:{Proven}\:{that}\:{by}\:{all}\:{n}\:\in\:{N}^{\ast} \\ $$$$\:\mathrm{2}!\mathrm{4}!..\left(\mathrm{2}{n}\right)!\geqslant\left(\left({n}+\mathrm{1}\right)!\right)^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{Proven}\:{by}\:{recurring}\:{that}\: \\ $$$$\sum_{{p}=\mathrm{1}} ^{{n}} {pp}!=\left({n}+\mathrm{1}\right)!−\mathrm{1} \\ $$

Question Number 158334    Answers: 1   Comments: 3

lim_(x→∞) (1/x)Σ_(r=1) ^x cos(((rπ)/(2x))) x∈N

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\underset{{r}=\mathrm{1}} {\overset{{x}} {\sum}}{cos}\left(\frac{{r}\pi}{\mathrm{2}{x}}\right) \\ $$$${x}\in\mathbb{N} \\ $$

Question Number 158333    Answers: 0   Comments: 0

∫_((1;π)) ^((2;π)) (1−(y^2 /x^2 )cos((y/x)))dx+(sin((y/x))+(y/x)cos((y/x)))dy=? OY on the road that doesn′t cut your arrow

$$\int_{\left(\mathrm{1};\pi\right)} ^{\left(\mathrm{2};\pi\right)} \left(\mathrm{1}−\frac{\boldsymbol{\mathrm{y}}^{\mathrm{2}} }{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\right)\right)\boldsymbol{\mathrm{dx}}+\left(\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\right)+\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\right)\right)\boldsymbol{\mathrm{dy}}=? \\ $$$$\boldsymbol{\mathrm{OY}}\:\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{road}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{doesn}}'\boldsymbol{\mathrm{t}}\:\boldsymbol{\mathrm{cut}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{arrow}} \\ $$

Question Number 158335    Answers: 2   Comments: 0

if α,β,γ are the angles of a triangle, find (1/(tan 𝛂 tan 𝛃))+(1/(tan 𝛃 tan 𝛄))+(1/(tan 𝛄 tan 𝛂))=?

$${if}\:\alpha,\beta,\gamma\:{are}\:{the}\:{angles}\:{of}\:{a}\:{triangle}, \\ $$$${find}\: \\ $$$$\frac{\mathrm{1}}{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\alpha}\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{\beta}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\beta}\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{\gamma}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\gamma}\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{\alpha}}=? \\ $$

Question Number 158331    Answers: 0   Comments: 0

Prove that: Σ_(n=1) ^∞ (1/( (√n) (n + 1))) < 2

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}}\:\left(\mathrm{n}\:+\:\mathrm{1}\right)}\:<\:\mathrm{2} \\ $$

Question Number 158330    Answers: 0   Comments: 1

Question Number 158327    Answers: 0   Comments: 0

Show that the sequence (x^n /(1 + x^n )) does not converge uniformly on [0, 2] by showing that the limit function is not continuous on [0, 2]

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\:\:\:\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{1}\:\:\:+\:\:\:\mathrm{x}^{\mathrm{n}} }\:\:\:\:\mathrm{does}\:\mathrm{not}\:\mathrm{converge}\:\mathrm{uniformly}\:\mathrm{on}\:\:\left[\mathrm{0},\:\:\mathrm{2}\right] \\ $$$$\mathrm{by}\:\mathrm{showing}\:\mathrm{that}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{function}\:\mathrm{is}\:\mathrm{not}\:\mathrm{continuous}\:\mathrm{on}\:\:\:\:\left[\mathrm{0},\:\:\mathrm{2}\right] \\ $$

Question Number 158325    Answers: 2   Comments: 0

Question Number 158322    Answers: 0   Comments: 0

prove that: Σ_(n=1) ^∞ tan^( −1) ( (1/F_( n) ) ).tan^( −1) ( (1/F_( n+1) ) )= (π^( 2) /8) Fibonacci numbers

$$ \\ $$$$\:\:{prove}\:\:{that}: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:{tan}^{\:−\mathrm{1}} \:\left(\:\frac{\mathrm{1}}{\mathrm{F}_{\:{n}} }\:\right).{tan}^{\:−\mathrm{1}} \left(\:\frac{\mathrm{1}}{\mathrm{F}_{\:{n}+\mathrm{1}} }\:\right)=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{8}} \\ $$$$\:\:\mathrm{F}{ibonacci}\:{numbers} \\ $$$$ \\ $$

Question Number 158320    Answers: 0   Comments: 0

prove that : Σ_(n=1) ^∞ (( H_( n) . F_n )/2^( n) ) = ln(4) + ((12)/( (√5))) ln( ϕ ) ϕ : Golden ratio F_( n) : fibonacci numbers

$$ \\ $$$$\:\:\:{prove}\:\:{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{H}_{\:{n}} .\:\mathrm{F}_{{n}} }{\mathrm{2}^{\:{n}} }\:\:=\:{ln}\left(\mathrm{4}\right)\:+\:\frac{\mathrm{12}}{\:\sqrt{\mathrm{5}}}\:{ln}\left(\:\varphi\:\right) \\ $$$$\:\:\:\:\:\varphi\::\:\:\:\mathrm{Golden}\:\:\mathrm{ratio} \\ $$$$\:\:\:\:\:\:\mathrm{F}_{\:{n}} \::\:{fibonacci}\:{numbers} \\ $$$$ \\ $$

Question Number 158721    Answers: 1   Comments: 0

Find: ∫_( 0) ^( ∞) (1/((x^2 + x + 1)(1 + ax))) dx ; a>0

$$\mathrm{Find}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{1}\right)\left(\mathrm{1}\:+\:\mathrm{ax}\right)}\:\mathrm{dx}\:\:;\:\:\mathrm{a}>\mathrm{0} \\ $$$$ \\ $$

Question Number 158313    Answers: 1   Comments: 0

Question Number 158309    Answers: 3   Comments: 0

y′′+y′−2y=−18te^(−2t)

$${y}''+{y}'−\mathrm{2}{y}=−\mathrm{18}{te}^{−\mathrm{2}{t}} \\ $$

Question Number 158306    Answers: 0   Comments: 3

Question Number 158366    Answers: 1   Comments: 0

Question Number 158303    Answers: 1   Comments: 0

x;y;z;t>0 solve for real numbers: { ((8x^4 + 64y^4 + 216z^4 + 1728t^4 = 1)),((x + y + z + t = 1)) :}

$$\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}>\mathrm{0} \\ $$$$\mathrm{solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\begin{cases}{\mathrm{8x}^{\mathrm{4}} \:+\:\mathrm{64y}^{\mathrm{4}} \:+\:\mathrm{216z}^{\mathrm{4}} \:+\:\mathrm{1728t}^{\mathrm{4}} \:=\:\mathrm{1}}\\{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:+\:\mathrm{t}\:=\:\mathrm{1}}\end{cases} \\ $$$$ \\ $$

Question Number 158425    Answers: 0   Comments: 0

Question Number 158424    Answers: 1   Comments: 0

Question Number 158301    Answers: 1   Comments: 1

proven that 1^0 =1 et que 0!=1

$${proven}\:{that}\: \\ $$$$\mathrm{1}^{\mathrm{0}} =\mathrm{1}\:{et}\:{que}\:\mathrm{0}!=\mathrm{1} \\ $$

Question Number 158295    Answers: 0   Comments: 0

𝛀 =∫_( 0) ^( 1) ((sin^(-1) x log(1 + x))/x^2 ) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{sin}^{-\mathrm{1}} \:\mathrm{x}\:\mathrm{log}\left(\mathrm{1}\:+\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$$$ \\ $$

Question Number 158285    Answers: 0   Comments: 1

Question Number 158293    Answers: 3   Comments: 0

question# If , Ω =∫_0 ^( 1) ((ln^( 2) (1−x^( 4) ))/x) dx= a ζ b) find the value of , a , b .

$$ \\ $$$$\:\:\:\:\:{question}# \\ $$$$\left.\mathrm{If}\:,\:\:\Omega\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}^{\:\mathrm{4}} \right)}{{x}}\:{dx}=\:{a}\:\zeta\:{b}\right) \\ $$$$\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}\:,\:\:\:\:\:{a}\:\:,\:{b}\:\:. \\ $$$$ \\ $$$$ \\ $$

Question Number 158317    Answers: 0   Comments: 0

Question Number 158290    Answers: 0   Comments: 0

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