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Question Number 120312 Answers: 1 Comments: 0

... ♣nice calculus♣... prove that :: lim_(n→∞) (((n$))^(1/n^2 ) /( (√n)))?=^(???) e^((−3)/4) where :: n$ =^(superfactorial) n!.(n−1)!.(n−2)!...3!.2!.1! ...♠m.n.1970♠...

$$\:\:\:\:\:\:\:\:\:\:...\:\clubsuit{nice}\:\:{calculus}\clubsuit... \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \frac{\sqrt[{{n}^{\mathrm{2}} }]{{n\$}}}{\:\sqrt{{n}}}?\overset{???} {=}{e}^{\frac{−\mathrm{3}}{\mathrm{4}}} \\ $$$$\:\:\:\:\:{where}\:::\:\:{n\$}\:\overset{{superfactorial}} {=}{n}!.\left({n}−\mathrm{1}\right)!.\left({n}−\mathrm{2}\right)!...\mathrm{3}!.\mathrm{2}!.\mathrm{1}! \\ $$$$\:\:\:\:\:\:\:\:...\spadesuit{m}.{n}.\mathrm{1970}\spadesuit... \\ $$

Question Number 120526 Answers: 0 Comments: 1

Question Number 120301 Answers: 1 Comments: 3

is it possible to have(1+(1/x^4 ))^(1/2) =(1+(1/(2x^4 ))) please help

$${is}\:{it}\:{possible}\:{to}\:{have}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\right)^{\frac{\mathrm{1}}{\mathrm{2}}} =\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{4}} }\right)\:{please}\:{help} \\ $$

Question Number 120300 Answers: 0 Comments: 1

Determine all function f:R→R which satisfy f(a+x)−f(a−x)=4ax for all real a and x.

$${Determine}\:{all}\:{function}\:{f}:{R}\rightarrow{R} \\ $$$${which}\:{satisfy}\:{f}\left({a}+{x}\right)−{f}\left({a}−{x}\right)=\mathrm{4}{ax} \\ $$$${for}\:{all}\:{real}\:{a}\:{and}\:{x}. \\ $$

Question Number 120297 Answers: 3 Comments: 0

∫_0 ^(π/2) ((x dx)/(sin x+cos x))

$$\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{{x}\:{dx}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}} \\ $$

Question Number 120295 Answers: 1 Comments: 1

Question Number 120288 Answers: 1 Comments: 1

let f(x)=((ln(1+x^2 ))/(x^2 +3)) determine f^((n)) (x) and developp f at integr serie

$${let}\:{f}\left({x}\right)=\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} +\mathrm{3}} \\ $$$${determine}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$${and}\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 120285 Answers: 0 Comments: 0

calculate ∫_2 ^∞ (dx/((x^2 −1)^2 (x^2 +x+1)))

$${calculate}\:\:\int_{\mathrm{2}} ^{\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)} \\ $$

Question Number 120284 Answers: 1 Comments: 1

Question Number 120283 Answers: 0 Comments: 0

fond ∫_2 ^∞ ((ln(1+3x^2 ))/(1+x^4 ))dx

$${fond}\:\int_{\mathrm{2}} ^{\infty} \frac{{ln}\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 120275 Answers: 1 Comments: 0

(((3+2 (5)^(1/4) )/(3−2 (5)^(1/4) )))^(1/(4 )) =?

$$\:\sqrt[{\mathrm{4}\:}]{\frac{\mathrm{3}+\mathrm{2}\:\sqrt[{\mathrm{4}}]{\mathrm{5}}}{\mathrm{3}−\mathrm{2}\:\sqrt[{\mathrm{4}}]{\mathrm{5}}}}\:=?\: \\ $$

Question Number 120274 Answers: 0 Comments: 0

Question Number 120272 Answers: 0 Comments: 0

ϕ(x)=x^r where x∈[0,+∞[ and 0<r≤1 shown that for any (x,y)∈R_+ ^2 𝛟(x+y)≤𝛟(x)+𝛟(y) please

$$\varphi\left({x}\right)=\boldsymbol{{x}}^{\boldsymbol{{r}}} \\ $$$$\boldsymbol{{where}}\:\boldsymbol{{x}}\in\left[\mathrm{0},+\infty\left[\:\boldsymbol{{and}}\:\mathrm{0}<\boldsymbol{{r}}\leqslant\mathrm{1}\right.\right. \\ $$$${shown}\:{that}\:{for}\:{any}\:\left(\boldsymbol{{x}},\boldsymbol{{y}}\right)\in\mathbb{R}_{+} ^{\mathrm{2}} \\ $$$$\boldsymbol{\varphi}\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\leqslant\boldsymbol{\varphi}\left(\boldsymbol{{x}}\right)+\boldsymbol{\varphi}\left(\boldsymbol{{y}}\right) \\ $$$$\:\:\:\:\boldsymbol{{please}} \\ $$

Question Number 120268 Answers: 3 Comments: 0

Question Number 120258 Answers: 1 Comments: 0

Question Number 120257 Answers: 2 Comments: 1

∫ ((f ′(x))/(f(x))) =?

$$\:\int\:\frac{{f}\:'\left({x}\right)}{{f}\left({x}\right)}\:=? \\ $$

Question Number 120254 Answers: 4 Comments: 0

∫ (dx/(1+cosθ.cos x )) ?

$$\:\int\:\frac{{dx}}{\mathrm{1}+\mathrm{cos}\theta.\mathrm{cos}\:{x}\:}\:? \\ $$

Question Number 120244 Answers: 3 Comments: 2

how to justify that sin (x−((7π)/2) )= cos x

$${how}\:{to}\:{justify}\:\:{that}\:\mathrm{sin}\:\left({x}−\frac{\mathrm{7}\pi}{\mathrm{2}}\:\right)=\:\mathrm{cos}\:{x} \\ $$

Question Number 120243 Answers: 1 Comments: 0

Question Number 120240 Answers: 1 Comments: 0

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Question Number 120241 Answers: 0 Comments: 0

Correction to the last assignment (1)(1/(81^(x−2) )) = 27^(1−x) (1/3^(4(x−2)) )=3^(3(1−x)) (1/3^(4x−8) )=3^(3−3x) 3^(−4x+8) =3^(3−3x) −4x+8=3−3x C.L.T −4x+3x=3−8 −x=−5 x=5 (2)9^x =(1/(729)) 9^x =(1/3^6 ) 3^(2x) =3^(−6) 2x=−6 x=((−6)/2) x=−3 (3)16^x =0.125 2^(4x) =((125)/(1000)) 2^(4x) =(1/8) 2^(4x) =2^(−3) 4x=−3 x=((−3)/4) (4)a^(1/2) =4 a=

$$\boldsymbol{\mathrm{C}}{orrection}\:\boldsymbol{{to}}\:\boldsymbol{{the}}\:\boldsymbol{{last}}\:\boldsymbol{{assignment}} \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\frac{\mathrm{1}}{\mathrm{81}^{\boldsymbol{{x}}−\mathrm{2}} }\:=\:\mathrm{27}^{\mathrm{1}−\boldsymbol{{x}}} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{4}\left(\boldsymbol{{x}}−\mathrm{2}\right)} }=\mathrm{3}^{\mathrm{3}\left(\mathrm{1}−\boldsymbol{{x}}\right)} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{4}\boldsymbol{{x}}−\mathrm{8}} }=\mathrm{3}^{\mathrm{3}−\mathrm{3}\boldsymbol{{x}}} \\ $$$$\mathrm{3}^{−\mathrm{4}\boldsymbol{{x}}+\mathrm{8}} =\mathrm{3}^{\mathrm{3}−\mathrm{3}\boldsymbol{{x}}} \\ $$$$−\mathrm{4}\boldsymbol{{x}}+\mathrm{8}=\mathrm{3}−\mathrm{3}\boldsymbol{{x}} \\ $$$$\boldsymbol{{C}}.{L}.{T} \\ $$$$−\mathrm{4}{x}+\mathrm{3}{x}=\mathrm{3}−\mathrm{8} \\ $$$$−{x}=−\mathrm{5} \\ $$$${x}=\mathrm{5} \\ $$$$ \\ $$$$\left(\mathrm{2}\right)\mathrm{9}^{{x}} =\frac{\mathrm{1}}{\mathrm{729}} \\ $$$$\mathrm{9}^{{x}} =\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{6}} } \\ $$$$\mathrm{3}^{\mathrm{2}{x}} =\mathrm{3}^{−\mathrm{6}} \\ $$$$\mathrm{2}{x}=−\mathrm{6} \\ $$$${x}=\frac{−\mathrm{6}}{\mathrm{2}} \\ $$$${x}=−\mathrm{3} \\ $$$$ \\ $$$$\left(\mathrm{3}\right)\mathrm{16}^{{x}} =\mathrm{0}.\mathrm{125} \\ $$$$\mathrm{2}^{\mathrm{4}{x}} =\frac{\mathrm{125}}{\mathrm{1000}} \\ $$$$\mathrm{2}^{\mathrm{4}{x}} =\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\mathrm{2}^{\mathrm{4}{x}} =\mathrm{2}^{−\mathrm{3}} \\ $$$$\mathrm{4}{x}=−\mathrm{3} \\ $$$${x}=\frac{−\mathrm{3}}{\mathrm{4}} \\ $$$$ \\ $$$$\left(\mathrm{4}\right){a}^{\mathrm{1}/\mathrm{2}} =\mathrm{4} \\ $$$${a}= \\ $$

Question Number 120238 Answers: 0 Comments: 0

Question Number 120233 Answers: 0 Comments: 0

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Question Number 120237 Answers: 0 Comments: 1

Question Number 120230 Answers: 3 Comments: 0

∫ (x^3 /((x +2)^4 )) dx OR ∫ ((cos x)/(1 + cos x)) dx a^→ = i^ − j^ + 3k^ and b^(→ ) = 2i^ − 7j^ + k^ .

$$\:\:\:\:\int\:\frac{\mathrm{x}^{\mathrm{3}} }{\left(\mathrm{x}\:+\mathrm{2}\right)^{\mathrm{4}} }\:\mathrm{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{OR} \\ $$$$\:\:\:\int\:\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{1}\:+\:\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx} \\ $$$$ \\ $$$$\:\:\:\:\:\overset{\rightarrow} {\mathrm{a}}\:=\:\hat {\mathrm{i}}\:−\:\hat {\mathrm{j}}\:+\:\mathrm{3}\hat {\mathrm{k}}\:\:\mathrm{and}\:\overset{\rightarrow\:} {\mathrm{b}}\:=\:\mathrm{2}\hat {\mathrm{i}}\:−\:\mathrm{7}\hat {\mathrm{j}}\:+\:\hat {\mathrm{k}}\:. \\ $$$$ \\ $$$$\:\:\:\: \\ $$

Question Number 120229 Answers: 2 Comments: 0

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