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Question Number 146076 Answers: 1 Comments: 0

if f(x)=∫(3x^2 −2)dx and f(2)=9 then f(−2)= ?

$${if}\:{f}\left({x}\right)=\int\left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}\right){dx}\:{and}\:{f}\left(\mathrm{2}\right)=\mathrm{9}\:{then} \\ $$$${f}\left(−\mathrm{2}\right)=\:? \\ $$

Question Number 146073 Answers: 0 Comments: 0

Let K be nonempty corps , K^∗ =K−{0_K } Prove that 1) Π_(x∈K^∗ ) x = −1 2)Deduce that p is prime ⇔ (p−1)!≡−1[p]

$${Let}\:{K}\:{be}\:{nonempty}\:\:{corps}\:,\:{K}^{\ast} ={K}−\left\{\mathrm{0}_{{K}} \right\} \\ $$$${Prove}\:{that} \\ $$$$\left.\mathrm{1}\right)\:\underset{{x}\in{K}^{\ast} } {\prod}{x}\:=\:−\mathrm{1} \\ $$$$\left.\mathrm{2}\right){Deduce}\:{that}\: \\ $$$$\:\:{p}\:{is}\:{prime}\:\Leftrightarrow\:\left({p}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[{p}\right] \\ $$

Question Number 146072 Answers: 0 Comments: 0

Let F_n =2^2^n +1 the fermat number Prove that F_n is prime ⇔ 3^((F_n −1)/2) ≡1[F_n ]

$${Let}\:{F}_{{n}} =\mathrm{2}^{\mathrm{2}^{{n}} } +\mathrm{1}\:{the}\:{fermat}\:{number} \\ $$$${Prove}\:{that} \\ $$$$\:{F}_{{n}} \:{is}\:{prime}\:\Leftrightarrow\:\mathrm{3}^{\frac{{F}_{{n}} −\mathrm{1}}{\mathrm{2}}} \equiv\mathrm{1}\left[{F}_{{n}} \right] \\ $$

Question Number 146067 Answers: 1 Comments: 0

transform the cartesian inyegral ∫_0 ^1 ∫_0 ^(√(1−x^2 )) e^(−(x^2 +y^2 )) dy dx into polar integral and evaluate it.

$${transform}\:{the}\:{cartesian}\:{inyegral}\: \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} {\int}}{e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} \:{dy}\:{dx}\:{into}\:{polar}\:{integral}\: \\ $$$${and}\:{evaluate}\:{it}. \\ $$

Question Number 146063 Answers: 1 Comments: 0

if g(x)=((x^( 2) −x)/(2x−1)) , D_g = [1 , ∞) , lim_(x→∞) ((g^( −1) (x))/(ax + b)) = b−a (a <0 ) then find the value of Max (b ) D_( g) = Domain

$$ \\ $$$$\:\:\:\:{if}\:\:{g}\left({x}\right)=\frac{{x}^{\:\mathrm{2}} −{x}}{\mathrm{2}{x}−\mathrm{1}}\:\:\:,\:{D}_{{g}} =\:\left[\mathrm{1}\:,\:\infty\right) \\ $$$$\:\:\:\:,\:{lim}_{{x}\rightarrow\infty} \frac{{g}^{\:−\mathrm{1}} \left({x}\right)}{{ax}\:+\:{b}}\:=\:{b}−{a}\:\:\left({a}\:<\mathrm{0}\:\right) \\ $$$$\:\:{then}\:{find}\:\:{the}\:{value}\:{of}\:{Max}\:\left({b}\:\right) \\ $$$$\:\: \\ $$$$\:\:{D}_{\:{g}} \:=\:{Domain}\: \\ $$

Question Number 146062 Answers: 0 Comments: 1

find values a , b , c such that: −1≤ ax^2 +bx +c ≤ 1 and ((6b^( 2) + 8 a^( 2) )/3) is Max...

$$ \\ $$$$\:\:\:\:{find}\:\:{values}\:\:{a}\:,\:{b}\:,\:{c}\:\:{such}\:{that}: \\ $$$$\:\:\:\:−\mathrm{1}\leqslant\:{ax}\:^{\mathrm{2}} +{bx}\:+{c}\:\leqslant\:\mathrm{1} \\ $$$$\:\:\:\:\:\:{and}\:\:\frac{\mathrm{6}{b}^{\:\mathrm{2}} +\:\mathrm{8}\:{a}^{\:\mathrm{2}} }{\mathrm{3}}\:{is}\:{Max}... \\ $$

Question Number 146061 Answers: 0 Comments: 0

Question Number 146057 Answers: 1 Comments: 0

Question Number 146054 Answers: 0 Comments: 0

I := ∫_0 ^( ∞) e^( −x) . J_(1/2) (x ) dx J_(v ) (x ) = x^( v) Σ_(n=0) ^( ∞) (((− 1 )^( n) x^( 2n) )/(2^( n + v) n ! Γ ( n + v +1 ))) ....

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\:\infty} {e}^{\:−{x}} \:.\:\mathrm{J}_{\frac{\mathrm{1}}{\mathrm{2}}} \:\left({x}\:\right)\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{J}_{{v}\:} \:\left({x}\:\right)\:=\:{x}^{\:{v}} \:\underset{{n}=\mathrm{0}} {\overset{\:\infty} {\sum}}\frac{\left(−\:\mathrm{1}\:\right)^{\:{n}} \:{x}^{\:\mathrm{2}{n}} }{\mathrm{2}^{\:{n}\:+\:{v}} \:{n}\:!\:\Gamma\:\left(\:{n}\:+\:{v}\:+\mathrm{1}\:\right)} \\ $$$$\:\:\:.... \\ $$

Question Number 146048 Answers: 1 Comments: 0

in a triangle ABC we have { ((3sinA^ +4cosB^ =6)),((4sinB^ +3cosA^ =1)) :} find C^

$${in}\:{a}\:{triangle}\:{ABC}\:\:{we}\:{have}\: \\ $$$$\begin{cases}{\mathrm{3}{sin}\hat {{A}}+\mathrm{4}{cos}\hat {{B}}=\mathrm{6}}\\{\mathrm{4}{sin}\hat {{B}}+\mathrm{3}{cos}\hat {{A}}=\mathrm{1}}\end{cases} \\ $$$${find}\:\hat {{C}} \\ $$$$ \\ $$

Question Number 146046 Answers: 1 Comments: 0

f(x) = x^(sin(x)) ⇒ f^′ (x) = ?

$${f}\left({x}\right)\:=\:{x}^{\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)} \:\Rightarrow\:{f}\:^{'} \left({x}\right)\:=\:? \\ $$

Question Number 146043 Answers: 2 Comments: 0

Question Number 146044 Answers: 1 Comments: 0

Simplify: ((sin^3 α)/(1-cosα)) + ((cos^3 α)/(sinα+1)) = ?

$${Simplify}: \\ $$$$\frac{{sin}^{\mathrm{3}} \alpha}{\mathrm{1}-{cos}\alpha}\:+\:\frac{{cos}^{\mathrm{3}} \alpha}{{sin}\alpha+\mathrm{1}}\:=\:? \\ $$

Question Number 146035 Answers: 2 Comments: 0

help me please ∫((ln(x+1))/x)dx=??

$${help}\:{me}\:{please} \\ $$$$\int\frac{{ln}\left({x}+\mathrm{1}\right)}{{x}}{dx}=?? \\ $$$$ \\ $$

Question Number 146030 Answers: 0 Comments: 1

Question Number 146026 Answers: 2 Comments: 3

Question Number 146017 Answers: 1 Comments: 0

Question Number 146013 Answers: 0 Comments: 0

Question Number 146009 Answers: 1 Comments: 0

(1/4)+(1/(12))+(1/(24))+...+(1/(2n(n+1)))=?

$$\:\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{12}}+\frac{\mathrm{1}}{\mathrm{24}}+...+\frac{\mathrm{1}}{\mathrm{2n}\left(\mathrm{n}+\mathrm{1}\right)}=? \\ $$

Question Number 146004 Answers: 1 Comments: 0

F et G deux sous espaces vectoriels de E a) montrer que (F∩G=F+G)⇔(F=G) b) quand dit−on que les deux sous espaces vectoriels F et G sont supplementaires?

$$\mathrm{F}\:\mathrm{et}\:\mathrm{G}\:\mathrm{deux}\:\mathrm{sous}\:\mathrm{espaces}\:\mathrm{vectoriels}\:\mathrm{de}\:\mathrm{E} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{montrer}\:\mathrm{que}\:\left(\mathrm{F}\cap\mathrm{G}=\mathrm{F}+\mathrm{G}\right)\Leftrightarrow\left(\mathrm{F}=\mathrm{G}\right) \\ $$$$\left.\mathrm{b}\right)\:\mathrm{quand}\:\mathrm{dit}−\mathrm{on}\:\mathrm{que}\:\mathrm{les}\:\mathrm{deux}\:\mathrm{sous}\:\mathrm{espaces}\: \\ $$$$\mathrm{vectoriels}\:\mathrm{F}\:\mathrm{et}\:\mathrm{G}\:\mathrm{sont}\:\mathrm{supplementaires}? \\ $$

Question Number 145996 Answers: 3 Comments: 0

find the sum of x−2x^2 +3x^3 −4x^4 +5x^5 −6x^6 +... where ∣x∣ < 1

$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\: \\ $$$$\:\mathrm{x}−\mathrm{2x}^{\mathrm{2}} +\mathrm{3x}^{\mathrm{3}} −\mathrm{4x}^{\mathrm{4}} +\mathrm{5x}^{\mathrm{5}} −\mathrm{6x}^{\mathrm{6}} +... \\ $$$$\mathrm{where}\:\mid\mathrm{x}\mid\:<\:\mathrm{1} \\ $$

Question Number 148416 Answers: 1 Comments: 0

lim_(x→∞) ((x! - cos(2x))/(3x + 1)) = ?

$$\underset{\boldsymbol{{x}}\rightarrow\infty} {{lim}}\frac{{x}!\:-\:{cos}\left(\mathrm{2}{x}\right)}{\mathrm{3}{x}\:+\:\mathrm{1}}\:=\:? \\ $$

Question Number 145986 Answers: 1 Comments: 0

What is x−f(x)×e/3.5π×θ=??????? function f(input) = decrypt( determinant ((( determinant (( )) )))

$${What}\:{is} \\ $$$${x}−{f}\left({x}\right)×{e}/\mathrm{3}.\mathrm{5}\pi×\theta=??????? \\ $$$${function}\:{f}\left({input}\right)\:=\:{decrypt}\left(\begin{array}{|c|}{\begin{array}{|c|}{\underbrace{ }}\\\hline\end{array}\underbrace{ }}\\\hline\end{array}\right. \\ $$

Question Number 145982 Answers: 1 Comments: 0

Σ_(n≥0) (−(1/(81)))^n Γ(3n+3)=??

$$\underset{{n}\geqslant\mathrm{0}} {\sum}\left(−\frac{\mathrm{1}}{\mathrm{81}}\right)^{{n}} \Gamma\left(\mathrm{3}{n}+\mathrm{3}\right)=?? \\ $$

Question Number 145981 Answers: 0 Comments: 0

let f(x) be a function period 2π such that:f(x)={x, 0<x<π {π, π<x<2π show that the fourier series for f(x) in the interval 0<x<2π is ((3π)/4)−(2/π)[cosx+(1/3^2 )cos3x+(1/5^2 )cos5x+...]−[sinx+(1/2)sin2x+(1/3)sin3x+...]

$${let}\:{f}\left({x}\right)\:{be}\:{a}\:{function}\:{period}\:\mathrm{2}\pi\:{such}\:{that}:{f}\left({x}\right)=\left\{{x},\:\mathrm{0}<{x}<\pi\right. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{\pi,\:\pi<{x}<\mathrm{2}\pi\right. \\ $$$${show}\:{that}\:{the}\:{fourier}\:{series}\:{for}\:{f}\left({x}\right)\:{in}\:{the}\:{interval}\:\mathrm{0}<{x}<\mathrm{2}\pi\:{is} \\ $$$$\frac{\mathrm{3}\pi}{\mathrm{4}}−\frac{\mathrm{2}}{\pi}\left[{cosx}+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }{cos}\mathrm{3}{x}+\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }{cos}\mathrm{5}{x}+...\right]−\left[{sinx}+\frac{\mathrm{1}}{\mathrm{2}}{sin}\mathrm{2}{x}+\frac{\mathrm{1}}{\mathrm{3}}{sin}\mathrm{3}{x}+...\right] \\ $$

Question Number 145979 Answers: 1 Comments: 0

sin^2 x−4cos^3 x−1=2cos^2 x+2cos x−2cos xsin^2 x x=?

$$\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{4cos}\:^{\mathrm{3}} \mathrm{x}−\mathrm{1}=\mathrm{2cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{2cos}\:\mathrm{x}−\mathrm{2cos}\:\mathrm{xsin}\:^{\mathrm{2}} \mathrm{x} \\ $$$$\mathrm{x}=? \\ $$

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