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Question Number 153214 Answers: 1 Comments: 0

let a,b∈N^∗ a∗b=a+b+ab a^((n)) =a^((n−1)) ∗a explicite a^((n)) en fonction de a

$$\boldsymbol{{let}}\:\boldsymbol{{a}},\boldsymbol{{b}}\in\mathbb{N}^{\ast} \: \\ $$$$\:\boldsymbol{{a}}\ast\boldsymbol{{b}}=\boldsymbol{{a}}+\boldsymbol{{b}}+\boldsymbol{{ab}} \\ $$$$\boldsymbol{{a}}^{\left(\boldsymbol{{n}}\right)} =\boldsymbol{{a}}^{\left(\boldsymbol{{n}}−\mathrm{1}\right)} \ast\boldsymbol{{a}} \\ $$$$\boldsymbol{{explicite}}\:\boldsymbol{{a}}^{\left(\boldsymbol{{n}}\right)} \:\boldsymbol{{en}}\:\boldsymbol{{fonction}}\:\boldsymbol{{de}}\:\boldsymbol{{a}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 153216 Answers: 0 Comments: 1

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Question Number 153212 Answers: 0 Comments: 0

∫_(−∞) ^( ∞) (( tan(x).Arctanh(cos(x)))/x)dx=?

$$ \\ $$$$\:\:\int_{−\infty} ^{\:\infty} \frac{\:{tan}\left({x}\right).{Arctanh}\left({cos}\left({x}\right)\right)}{{x}}{dx}=? \\ $$$$ \\ $$

Question Number 153208 Answers: 0 Comments: 0

Question Number 153201 Answers: 0 Comments: 0

Question Number 153200 Answers: 0 Comments: 0

∫_0 ^(e−1) ∫_0 ^(e−x−1) ∫_0 ^(x+y+e) ((ln(z−x−y))/((x−e)(x+y−e)))dxdydz=?

$$\int_{\mathrm{0}} ^{{e}−\mathrm{1}} \int_{\mathrm{0}} ^{{e}−{x}−\mathrm{1}} \int_{\mathrm{0}} ^{{x}+{y}+{e}} \frac{{ln}\left({z}−{x}−{y}\right)}{\left({x}−{e}\right)\left({x}+{y}−{e}\right)}{dxdydz}=? \\ $$

Question Number 153203 Answers: 2 Comments: 0

Question Number 153194 Answers: 0 Comments: 0

Question Number 153189 Answers: 2 Comments: 0

Question Number 153181 Answers: 0 Comments: 2

Question Number 153182 Answers: 0 Comments: 2

Question Number 153174 Answers: 0 Comments: 1

Question Number 153173 Answers: 2 Comments: 1

Question Number 153172 Answers: 1 Comments: 0

lim_ _(x→+oo) Σ_(k=o) ^n^2 (n/( n^2 +k^2 ))

$$\underset{{x}\rightarrow+{oo}} {\mathrm{li}\underset{} {{m}}}\underset{{k}={o}} {\overset{{n}^{\mathrm{2}} } {\sum}}\frac{{n}}{\:{n}^{\mathrm{2}} +{k}^{\mathrm{2}} } \\ $$

Question Number 153168 Answers: 1 Comments: 0

Question Number 153164 Answers: 1 Comments: 0

∫_(−1) ^( 1) ln(𝚪(x)) dx

$$\int_{−\mathrm{1}} ^{\:\mathrm{1}} \boldsymbol{{ln}}\left(\boldsymbol{\Gamma}\left(\boldsymbol{{x}}\right)\right)\:\boldsymbol{{dx}} \\ $$

Question Number 153162 Answers: 1 Comments: 0

Determine all the derivable functions f:R→R which satisfy ((f(x^3 ) - f(0))/(f(x) - f(0))) = x^2 ; ∀x≠0

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{the}\:\mathrm{derivable}\:\mathrm{functions} \\ $$$$\mathrm{f}:\mathbb{R}\rightarrow\mathbb{R}\:\:\mathrm{which}\:\mathrm{satisfy} \\ $$$$\frac{\mathrm{f}\left(\mathrm{x}^{\mathrm{3}} \right)\:-\:\mathrm{f}\left(\mathrm{0}\right)}{\mathrm{f}\left(\mathrm{x}\right)\:-\:\mathrm{f}\left(\mathrm{0}\right)}\:=\:\mathrm{x}^{\mathrm{2}} \:\:;\:\:\forall\mathrm{x}\neq\mathrm{0} \\ $$

Question Number 153161 Answers: 1 Comments: 0

If P(x)=aX^3 +bX+c ∈ Q[X] ; (a≠0) has the roots x_1 , x_2 and x_3 such that x_1 = x_2 x_3 then prove that x_1 = 0 if and only if b=c

$$\mathrm{If}\:\:\mathrm{P}\left(\mathrm{x}\right)=\mathrm{aX}^{\mathrm{3}} +\mathrm{bX}+{c}\:\in\:\mathrm{Q}\left[\mathrm{X}\right]\:\:;\:\:\left(\mathrm{a}\neq\mathrm{0}\right) \\ $$$$\mathrm{has}\:\mathrm{the}\:\mathrm{roots}\:\:\mathrm{x}_{\mathrm{1}} \:,\:\mathrm{x}_{\mathrm{2}} \:\:\mathrm{and}\:\:\mathrm{x}_{\mathrm{3}} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\mathrm{x}_{\mathrm{1}} \:=\:\mathrm{x}_{\mathrm{2}} \mathrm{x}_{\mathrm{3}} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{x}_{\mathrm{1}} \:=\:\mathrm{0}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:\mathrm{b}=\mathrm{c} \\ $$

Question Number 153160 Answers: 0 Comments: 0

Find the set: { x ∣ x = ((b∙(a + b))/(a^2 + b^2 )) ; a>0 , b>0}

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}: \\ $$$$\left\{\:\mathrm{x}\:\mid\:\mathrm{x}\:=\:\frac{\mathrm{b}\centerdot\left(\mathrm{a}\:+\:\mathrm{b}\right)}{\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} }\:\:;\:\:\mathrm{a}>\mathrm{0}\:,\:\mathrm{b}>\mathrm{0}\right\} \\ $$

Question Number 153154 Answers: 1 Comments: 0

lim_(x→y) ((sin (e^x )−sin (e^y ))/(x−y))=?

$$\:\:\underset{{x}\rightarrow{y}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left({e}^{{x}} \right)−\mathrm{sin}\:\left({e}^{{y}} \right)}{{x}−{y}}=? \\ $$

Question Number 153153 Answers: 1 Comments: 0

Question Number 153152 Answers: 2 Comments: 0

∫ (√(x^3 +1)) dx

$$\int\:\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx} \\ $$

Question Number 153151 Answers: 2 Comments: 0

∫_0 ^1 ((ln(x^3 +1))/(x+1))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}\left({x}^{\mathrm{3}} +\mathrm{1}\right)}{{x}+\mathrm{1}}{dx} \\ $$

Question Number 153146 Answers: 1 Comments: 0

Question Number 153128 Answers: 1 Comments: 0

if ((x+y)/2) + (√(xy)) = ((1+(√5))/2) and x^2 +y^2 =(√5) find (1/x) + (1/y) = ?

$$\mathrm{if}\:\:\:\frac{\mathrm{x}+\mathrm{y}}{\mathrm{2}}\:+\:\sqrt{\mathrm{xy}}\:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\sqrt{\mathrm{5}} \\ $$$$\mathrm{find}\:\:\:\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\:=\:? \\ $$

Question Number 153127 Answers: 1 Comments: 2

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