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Question Number 158869 Answers: 0 Comments: 2

The roots of the equation 2x^2 +px+p=0 are 2α+β and α+2β. Calculate the value of p

$$\:{The}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\:\mathrm{2}{x}^{\mathrm{2}} +{px}+{p}=\mathrm{0}\:{are}\:\mathrm{2}\alpha+\beta\:{and} \\ $$$$\:\alpha+\mathrm{2}\beta.\:{Calculate}\:{the}\:{value}\:{of}\:{p} \\ $$

Question Number 158862 Answers: 1 Comments: 0

Question Number 158863 Answers: 0 Comments: 0

Question Number 158858 Answers: 0 Comments: 0

I_n =∫_(−1) ^1 (1−x^2 )^n cos ((a/(2b))x)dx to integrating by piece for n≥2 proven (a^2 /(4b^2 ))I_(n ) =2n(2n−1)I_(n−1) −4(n−1)I_(n−2) proven by rearring that ((a/(2b)))^(2n+1) I_n =n![p((q/(2b)))sin ((a/(2b)))+Q((a/(2b)))cos ((a/(2b)))]

$${I}_{{n}} =\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} \mathrm{cos}\:\left(\frac{{a}}{\mathrm{2}{b}}{x}\right){dx} \\ $$$${to}\:{integrating}\:{by}\:{piece}\:{for}\:{n}\geqslant\mathrm{2}\: \\ $$$${proven}\: \\ $$$$\frac{{a}^{\mathrm{2}} }{\mathrm{4}{b}^{\mathrm{2}} }{I}_{{n}\:} =\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right){I}_{{n}−\mathrm{1}} −\mathrm{4}\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}} \\ $$$${proven}\:{by}\:{rearring}\:{that}\: \\ $$$$\left(\frac{{a}}{\mathrm{2}{b}}\right)^{\mathrm{2}{n}+\mathrm{1}} {I}_{{n}} ={n}!\left[{p}\left(\frac{{q}}{\mathrm{2}{b}}\right)\mathrm{sin}\:\left(\frac{{a}}{\mathrm{2}{b}}\right)+{Q}\left(\frac{{a}}{\mathrm{2}{b}}\right)\mathrm{cos}\:\left(\frac{{a}}{\mathrm{2}{b}}\right)\right] \\ $$

Question Number 158855 Answers: 1 Comments: 0

Resolve the system d′ unknow (x, y,z) ∈ ⊂^3 x+y+z=1 x^2 +y^2 +z^2 =1 x^3 +y^3 +z^3 =−5

$${Resolve}\:{the}\:{system}\:{d}'\:{unknow}\:\:\left({x},\:{y},{z}\right)\:\in\:\subset^{\mathrm{3}} \\ $$$${x}+{y}+{z}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{1} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =−\mathrm{5} \\ $$

Question Number 158847 Answers: 1 Comments: 0

Question Number 158843 Answers: 2 Comments: 2

Question Number 158839 Answers: 2 Comments: 0

Evaluate the following integrals using integration By Parts 1. ∫_((π )/4) ^(π/2) xcsc^2 xdx 2. ∫_1 ^(√3) arctan((1/x))dx

$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{following}\:\mathrm{integrals}\:\mathrm{using} \\ $$$$\mathrm{integration}\:\boldsymbol{\mathrm{By}}\:\boldsymbol{\mathrm{Parts}} \\ $$$$\mathrm{1}.\:\int_{\frac{\pi\:}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{2}}} {xcsc}^{\mathrm{2}} {xdx} \\ $$$$ \\ $$$$\mathrm{2}.\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} {arctan}\left(\frac{\mathrm{1}}{{x}}\right){dx} \\ $$

Question Number 158823 Answers: 0 Comments: 0

Question Number 158822 Answers: 1 Comments: 0

Σ_(n=0) ^∞ arctan (((−1)^n )/(2n+1))=?

$$\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\mathrm{arctan}\:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{2n}+\mathrm{1}}=? \\ $$

Question Number 158833 Answers: 1 Comments: 0

what is 1(1/2)%

$${what}\:{is}\:\mathrm{1}\frac{\mathrm{1}}{\mathrm{2}}\% \\ $$

Question Number 158829 Answers: 1 Comments: 2

The roots of the equation 2x^2 +px+q=0 are 2α+β and α+2β. Calculate the values of p and q

$$\:{The}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\:\mathrm{2}{x}^{\mathrm{2}} +{px}+{q}=\mathrm{0}\:{are}\:\mathrm{2}\alpha+\beta\:{and} \\ $$$$\:\alpha+\mathrm{2}\beta.\:{Calculate}\:{the}\:{values}\:{of} \\ $$$$\:{p}\:{and}\:{q} \\ $$

Question Number 158827 Answers: 1 Comments: 0

resolve ∫ln (cos x)dx

$${resolve}\:\int\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right){dx} \\ $$

Question Number 158816 Answers: 0 Comments: 0

Prove that 2017^(2017) and 2017^(2018) can be written as the sum of two perfect squares.

$$\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{2017}^{\mathrm{2017}} \:\:\mathrm{and}\:\:\:\mathrm{2017}^{\mathrm{2018}} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{perfect}\:\mathrm{squares}. \\ $$$$ \\ $$

Question Number 158814 Answers: 2 Comments: 0

Compare it: (log_4 20)^2 and log_4 320

$$\mathrm{Compare}\:\mathrm{it}: \\ $$$$\left(\mathrm{log}_{\mathrm{4}} \mathrm{20}\right)^{\mathrm{2}} \:\:\:\mathrm{and}\:\:\:\mathrm{log}_{\mathrm{4}} \mathrm{320} \\ $$$$ \\ $$

Question Number 158813 Answers: 0 Comments: 0

Find all value 𝛃>0 such that: ∫_( 0) ^( +∞) (dx/(x^(2021𝛃) + ln∙(1 + βx))) dx < +∞

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{value}\:\:\boldsymbol{\beta}>\mathrm{0}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:+\infty} {\int}}\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2021}\boldsymbol{\beta}} \:+\:\mathrm{ln}\centerdot\left(\mathrm{1}\:+\:\beta\mathrm{x}\right)}\:\mathrm{dx}\:<\:+\infty \\ $$

Question Number 158812 Answers: 0 Comments: 0

Question Number 158805 Answers: 0 Comments: 0

(1)F(x)= x^3 [ x ] ⇒ { ((F ′(0)=?)),((F ′(1)=?)) :} (2) F(x)= [ x ]−∣x∣ ⇒F ′(−(5/2))=? where [ ] : floor function ∣ ∣ absolute function

$$\left(\mathrm{1}\right){F}\left({x}\right)=\:{x}^{\mathrm{3}} \:\left[\:{x}\:\right]\:\Rightarrow\begin{cases}{{F}\:'\left(\mathrm{0}\right)=?}\\{{F}\:'\left(\mathrm{1}\right)=?}\end{cases} \\ $$$$\:\left(\mathrm{2}\right)\:{F}\left({x}\right)=\:\left[\:{x}\:\right]−\mid{x}\mid\:\Rightarrow{F}\:'\left(−\frac{\mathrm{5}}{\mathrm{2}}\right)=? \\ $$$$\:{where}\:\left[\:\right]\::\:{floor}\:{function} \\ $$$$\:\mid\:\mid\:{absolute}\:{function}\: \\ $$

Question Number 158803 Answers: 1 Comments: 1

Question Number 158794 Answers: 2 Comments: 0

montrer que 7 divise 2222^(5555) +5555^(2222)

$${montrer}\:{que}\:\mathrm{7}\:{divise} \\ $$$$\mathrm{2222}^{\mathrm{5555}} +\mathrm{5555}^{\mathrm{2222}} \\ $$$$ \\ $$

Question Number 158774 Answers: 1 Comments: 0

Question Number 158775 Answers: 1 Comments: 0

Find: ∫_( 0) ^( ∞) (1/((x^2 + x + 1)∙(1 + ax))) dx ; a>0 Answer: ((-π(√3)∙(a-2)+9a∙ln(a))/(9∙(a^2 -a+1)))

$$\mathrm{Find}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{1}\right)\centerdot\left(\mathrm{1}\:+\:\mathrm{ax}\right)}\:\mathrm{dx}\:\:;\:\:\mathrm{a}>\mathrm{0} \\ $$$$ \\ $$$$\mathrm{Answer}: \\ $$$$\frac{-\pi\sqrt{\mathrm{3}}\centerdot\left(\mathrm{a}-\mathrm{2}\right)+\mathrm{9a}\centerdot\mathrm{ln}\left(\mathrm{a}\right)}{\mathrm{9}\centerdot\left(\mathrm{a}^{\mathrm{2}} -\mathrm{a}+\mathrm{1}\right)} \\ $$

Question Number 158768 Answers: 1 Comments: 0

evaluate ∫2x(√(4x−5)) dx

$$ \\ $$$$\mathrm{evaluate} \\ $$$$\int\mathrm{2x}\sqrt{\mathrm{4x}−\mathrm{5}}\:\mathrm{dx} \\ $$

Question Number 158761 Answers: 0 Comments: 0

Question Number 158760 Answers: 1 Comments: 2

f(x)=[sgn(x^2 −1)+sgn(sin πx)] faind lim_(x→1) f(x)=?

$${f}\left({x}\right)=\left[{sgn}\left({x}^{\mathrm{2}} −\mathrm{1}\right)+{sgn}\left(\mathrm{sin}\:\pi{x}\right)\right] \\ $$$${faind}\:\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{f}\left({x}\right)=? \\ $$

Question Number 158759 Answers: 1 Comments: 0

Prove that: lim_(n→∞) ((Σ_(k=0) ^(2n) (-1)^k ∙ ((4n + 1)/(4n - 2k + 1)) (((2n)),(( k)) )))^(1/n) = 1

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\sqrt[{\boldsymbol{\mathrm{n}}}]{\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\mathrm{2}\boldsymbol{\mathrm{n}}} {\sum}}\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{k}}} \:\centerdot\:\frac{\mathrm{4n}\:+\:\mathrm{1}}{\mathrm{4n}\:-\:\mathrm{2k}\:+\:\mathrm{1}}\begin{pmatrix}{\mathrm{2n}}\\{\:\mathrm{k}}\end{pmatrix}}\:=\:\mathrm{1} \\ $$$$ \\ $$

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