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Question Number 159578 Answers: 0 Comments: 0

Find: 𝛀 =lim_(n→∞) ((((log(1 + (1/(n + 1))))^2 )/(log(1 + (1/(n + 2)))))) Answer: 0

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\left(\mathrm{log}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{n}\:+\:\mathrm{1}}\right)\right)^{\mathrm{2}} }{\mathrm{log}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{n}\:+\:\mathrm{2}}\right)}\right) \\ $$$$\mathrm{Answer}:\:\:\mathrm{0} \\ $$

Question Number 159568 Answers: 1 Comments: 1

Find: 𝛀 = ∫ sin^2 (x) ∙ cos(x) dx

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\:\int\:\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)\:\centerdot\:\mathrm{cos}\left(\mathrm{x}\right)\:\mathrm{dx} \\ $$$$ \\ $$

Question Number 159556 Answers: 2 Comments: 0

prove that: 2∤ a ⇒ 240∣ a^( 5) − a

$$ \\ $$$$\:\:{prove}\:{that}: \\ $$$$ \\ $$$$\:\:\:\:\mathrm{2}\nmid\:{a}\:\Rightarrow\:\mathrm{240}\mid\:{a}^{\:\mathrm{5}} \:−\:{a}\:\:\:\:\: \\ $$$$ \\ $$

Question Number 159552 Answers: 1 Comments: 4

Question Number 159551 Answers: 0 Comments: 0

hi ! help me for this one : lim_(x→0_(>) ) x E ((𝛑/x)) = ?

$$\boldsymbol{\mathrm{hi}}\:! \\ $$$$\boldsymbol{\mathrm{help}}\:\boldsymbol{\mathrm{me}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{one}}\:: \\ $$$$\:\:\:\:\:\underset{\underset{>} {\boldsymbol{{x}}\rightarrow\mathrm{0}}} {\boldsymbol{{lim}}}\:\boldsymbol{{x}}\:\boldsymbol{\mathrm{E}}\:\left(\frac{\boldsymbol{\pi}}{\boldsymbol{{x}}}\right)\:=\:?\: \\ $$

Question Number 159549 Answers: 1 Comments: 0

Question Number 159548 Answers: 0 Comments: 0

Question Number 159561 Answers: 0 Comments: 0

Question Number 159560 Answers: 1 Comments: 0

Resolve 1. u_(n+2) −2u_(n+1) +4u_n =3^n with u_o =1, u_1 =−2 2. u_n =u_(n−1) −u_(n−2) +2sin (((nΠ)/3)) with u_o =1, u_1 =2

$${Resolve}\: \\ $$$$\mathrm{1}.\:{u}_{{n}+\mathrm{2}} −\mathrm{2}{u}_{{n}+\mathrm{1}} +\mathrm{4}{u}_{{n}} =\mathrm{3}^{{n}} \\ $$$${with}\:{u}_{{o}} =\mathrm{1},\:{u}_{\mathrm{1}} =−\mathrm{2} \\ $$$$\mathrm{2}.\:{u}_{{n}} ={u}_{{n}−\mathrm{1}} −{u}_{{n}−\mathrm{2}} +\mathrm{2sin}\:\left(\frac{{n}\Pi}{\mathrm{3}}\right) \\ $$$${with}\:{u}_{{o}} =\mathrm{1},\:{u}_{\mathrm{1}} =\mathrm{2} \\ $$

Question Number 159540 Answers: 1 Comments: 0

prove that : 𝛗 := ∫_0 ^( ∞) (( sin((√x) ).sin((π/3) +(√x) ).sin(((2π)/3)+(√x) ).ln((1/x^( 2) ) ))/x)dx=^? π.(γ + ln(3) ) −−−−−−−−−− m.n

$$ \\ $$$$\:\: \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}\::=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left(\sqrt{{x}}\:\right).{sin}\left(\frac{\pi}{\mathrm{3}}\:+\sqrt{{x}}\:\right).{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}+\sqrt{{x}}\:\right).{ln}\left(\frac{\mathrm{1}}{{x}^{\:\mathrm{2}} }\:\right)}{{x}}{dx}\overset{?} {=}\:\pi.\left(\gamma\:+\:{ln}\left(\mathrm{3}\right)\:\right)\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:−−−−−−−−−−\:\:\:{m}.{n} \\ $$$$ \\ $$

Question Number 159534 Answers: 1 Comments: 0

Find: 𝛀 =∫_( 0) ^( 1) arctan^2 (x) dx

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{arctan}^{\mathrm{2}} \:\left(\mathrm{x}\right)\:\mathrm{dx} \\ $$$$ \\ $$

Question Number 159532 Answers: 0 Comments: 0

if 0<a≤b then: ∫_( a) ^( b) (x^(19) /( (√(1 + x^(30) )))) dx ≥ log (((2 + b^(20) )/(2 + a^(20) )))^(1/(10))

$$\mathrm{if}\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b}\:\:\mathrm{then}: \\ $$$$\underset{\:\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\:\frac{\mathrm{x}^{\mathrm{19}} }{\:\sqrt{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{30}} }}\:\mathrm{dx}\:\geqslant\:\mathrm{log}\:\sqrt[{\mathrm{10}}]{\frac{\mathrm{2}\:+\:\mathrm{b}^{\mathrm{20}} }{\mathrm{2}\:+\:\mathrm{a}^{\mathrm{20}} }} \\ $$$$ \\ $$

Question Number 159530 Answers: 0 Comments: 0

Question Number 159529 Answers: 1 Comments: 0

Solve for real numbers: { ((2x^2 + 3y^2 + z^2 = 7)),((x^2 + y^2 + z^2 = (√2) z (x + y))) :}

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\begin{cases}{\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{3y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{7}}\\{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\sqrt{\mathrm{2}}\:\mathrm{z}\:\left(\mathrm{x}\:+\:\mathrm{y}\right)}\end{cases} \\ $$$$ \\ $$

Question Number 159528 Answers: 1 Comments: 0

Find: 𝛀 =∫_( 0) ^( (𝛑/6)) ((sin(x)∙sin(x + (π/3))∙sin(x + ((2π)/3)))/(sin(3x) + cos(3x))) dx Answer: (π/(48))

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{6}}} {\int}}\frac{\mathrm{sin}\left(\mathrm{x}\right)\centerdot\mathrm{sin}\left(\mathrm{x}\:+\:\frac{\pi}{\mathrm{3}}\right)\centerdot\mathrm{sin}\left(\mathrm{x}\:+\:\frac{\mathrm{2}\pi}{\mathrm{3}}\right)}{\mathrm{sin}\left(\mathrm{3x}\right)\:+\:\mathrm{cos}\left(\mathrm{3x}\right)}\:\mathrm{dx} \\ $$$$\mathrm{Answer}:\:\:\frac{\pi}{\mathrm{48}} \\ $$

Question Number 159527 Answers: 1 Comments: 2

Question Number 159526 Answers: 0 Comments: 0

Ω= ∫_0 ^( ∞) (( sin^( 3) (x)ln(x))/x) dx=^(??) (π/8) (ln(3)−2γ) −−−−− solution.. Ω=∫_0^ ^( ∞) {(((3/4) sin(x)−(1/4) sin(3x))/x)} ln(x)dx = (3/4) (((−πγ)/2))− (1/4){ ∫_0 ^( ∞) ((sin(3x)ln(x))/x)dx=Ψ} ∴ Ψ := ∫_0 ^( ∞) (( sin(x).[ln(x)−ln(3)])/x)dx := −((πγ)/2) − ((ln(3).π)/2) ∴ Ω := ((−3πγ)/8) +((πγ)/8) +((π.ln(3))/8) := ((−2πγ)/8) + ((π.ln(3))/8) = (π/8) ( ln(3)−2γ )

$$ \\ $$$$\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{3}} \left({x}\right){ln}\left({x}\right)}{{x}}\:{dx}\overset{??} {=}\frac{\pi}{\mathrm{8}}\:\left({ln}\left(\mathrm{3}\right)−\mathrm{2}\gamma\right) \\ $$$$−−−−− \\ $$$$\:\:\:\:\:\:{solution}.. \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{0}^{\:} } ^{\:\infty} \left\{\frac{\frac{\mathrm{3}}{\mathrm{4}}\:{sin}\left({x}\right)−\frac{\mathrm{1}}{\mathrm{4}}\:{sin}\left(\mathrm{3}{x}\right)}{{x}}\right\}\:{ln}\left({x}\right){dx} \\ $$$$\:\:=\:\:\frac{\mathrm{3}}{\mathrm{4}}\:\left(\frac{−\pi\gamma}{\mathrm{2}}\right)−\:\frac{\mathrm{1}}{\mathrm{4}}\left\{\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\mathrm{3}{x}\right){ln}\left({x}\right)}{{x}}{dx}=\Psi\right\} \\ $$$$\:\:\therefore\:\:\Psi\::=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left({x}\right).\left[{ln}\left({x}\right)−{ln}\left(\mathrm{3}\right)\right]}{{x}}{dx} \\ $$$$\:\:\:\:\:\::=\:−\frac{\pi\gamma}{\mathrm{2}}\:\:−\:\frac{{ln}\left(\mathrm{3}\right).\pi}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\therefore\:\:\:\:\:\:\:\Omega\::=\:\:\frac{−\mathrm{3}\pi\gamma}{\mathrm{8}}\:+\frac{\pi\gamma}{\mathrm{8}}\:\:+\frac{\pi.{ln}\left(\mathrm{3}\right)}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\::=\:\frac{−\mathrm{2}\pi\gamma}{\mathrm{8}}\:+\:\frac{\pi.{ln}\left(\mathrm{3}\right)}{\mathrm{8}}\:=\:\frac{\pi}{\mathrm{8}}\:\left(\:{ln}\left(\mathrm{3}\right)−\mathrm{2}\gamma\:\right) \\ $$$$ \\ $$

Question Number 159525 Answers: 0 Comments: 0

I:=∫_0 ^( ∞) ((( sin^( 3) (x))/x^( 3) )) ln(x)dx=?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{I}:=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\:{sin}^{\:\mathrm{3}} \left({x}\right)}{{x}^{\:\mathrm{3}} }\right)\:{ln}\left({x}\right){dx}=? \\ $$$$ \\ $$

Question Number 159522 Answers: 0 Comments: 1

lim_(x→0) ((sin(sinx))/x)=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\left({sinx}\right)}{{x}}=? \\ $$

Question Number 159517 Answers: 1 Comments: 1

lim_(x→0) (1/x^2 )(e^(−2x) −((1+ax)/(1+bx))) a+b=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\left({e}^{−\mathrm{2}{x}} −\frac{\mathrm{1}+{ax}}{\mathrm{1}+{bx}}\right) \\ $$$${a}+{b}=? \\ $$

Question Number 159507 Answers: 2 Comments: 1