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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

Question Number 153117 Answers: 1 Comments: 0

∫_0 ^1 cot^(−1) (1−x+x^2 )dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{cot}^{−\mathrm{1}} \left(\mathrm{1}−\mathrm{x}+\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx} \\ $$

Question Number 153114 Answers: 2 Comments: 0

∫_0 ^1 ((log (1+x))/(1+x^2 ))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{log}\:\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 153112 Answers: 1 Comments: 0

∫_(−π) ^π (sin^(75) x+x^(125) )dx=0

$$\int_{−\pi} ^{\pi} \left(\mathrm{sin}\:^{\mathrm{75}} \mathrm{x}+\mathrm{x}^{\mathrm{125}} \right)\mathrm{dx}=\mathrm{0} \\ $$

Question Number 153111 Answers: 0 Comments: 0

∫_(−1) ^1 e^x dx as limit of the sum

$$\int_{−\mathrm{1}} ^{\mathrm{1}} \mathrm{e}^{\mathrm{x}} \:\mathrm{dx}\:\mathrm{as}\:\mathrm{limit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sum} \\ $$

Question Number 153110 Answers: 0 Comments: 0

∫_0 ^1 (3x^2 +2x+1)dx as the limit of sum

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{3x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{1}\right)\mathrm{dx}\: \\ $$$$\mathrm{as}\:\mathrm{the}\:\mathrm{limit}\:\:\mathrm{of}\:\mathrm{sum} \\ $$

Question Number 153209 Answers: 0 Comments: 2

Question Number 153104 Answers: 1 Comments: 0

∫_0 ^π (x^3 /(x^3 +(π−x)^3 ))dx=?

$$\int_{\mathrm{0}} ^{\pi} \frac{{x}^{\mathrm{3}} }{{x}^{\mathrm{3}} +\left(\pi−{x}\right)^{\mathrm{3}} }{dx}=? \\ $$

Question Number 153105 Answers: 1 Comments: 0

find ln 𝚪(x) ?

$${find}\:\:\boldsymbol{{ln}}\:\boldsymbol{\Gamma}\left(\boldsymbol{{x}}\right)\:? \\ $$

Question Number 153103 Answers: 1 Comments: 0

(1/(2∙3))+(1/(4∙5∙6))+(1/(7∙8∙9))+…

$$\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}\centerdot\mathrm{5}\centerdot\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{7}\centerdot\mathrm{8}\centerdot\mathrm{9}}+\ldots \\ $$

Question Number 153082 Answers: 1 Comments: 0

Question Number 153079 Answers: 1 Comments: 1

Solve the equation: (√(1-z)) = 2z^2 - 1 + 2z (√(1-z^2 ))

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\sqrt{\mathrm{1}-\boldsymbol{{z}}}\:=\:\mathrm{2}\boldsymbol{{z}}^{\mathrm{2}} \:-\:\mathrm{1}\:+\:\mathrm{2}\boldsymbol{{z}}\:\sqrt{\mathrm{1}-\boldsymbol{{z}}^{\mathrm{2}} } \\ $$

Question Number 153070 Answers: 3 Comments: 0

Find the solution set of inequality (√(3x+4)) ≥ x−2

$$\:\:\:{Find}\:{the}\:{solution}\:{set}\:{of}\:{inequality} \\ $$$$\:\:\:\:\:\:\:\sqrt{\mathrm{3}{x}+\mathrm{4}}\:\geqslant\:{x}−\mathrm{2}\: \\ $$

Question Number 153057 Answers: 1 Comments: 0

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