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Question Number 26772 Answers: 1 Comments: 0

With a center on a given circle of radius r ,an arc has been drawn in order to divide the circle in two equal (in area) parts. What is the radius of the arc in terms of r (radius of given circle)?

$$\mathrm{With}\:\mathrm{a}\:\mathrm{center}\:\boldsymbol{\mathrm{on}}\:\mathrm{a}\:\mathrm{given}\:\mathrm{circle}\:\mathrm{of} \\ $$$$\mathrm{radius}\:\mathrm{r}\:,\mathrm{an}\:\mathrm{arc}\:\mathrm{has}\:\mathrm{been}\:\mathrm{drawn}\:\mathrm{in}\:\mathrm{order} \\ $$$$\mathrm{to}\:\mathrm{divide}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{two}\:\mathrm{equal} \\ $$$$\left(\mathrm{in}\:\mathrm{area}\right)\:\mathrm{parts}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arc}\:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:\:\mathrm{r}\:\left(\mathrm{radius}\:\mathrm{of}\:\mathrm{given}\:\mathrm{circle}\right)?\: \\ $$

Question Number 26771 Answers: 0 Comments: 1

∫_( 0) ^( [x]) (2^x /2^([x]) ) dx =

$$\underset{\:\mathrm{0}} {\overset{\:\:\:\left[{x}\right]} {\int}}\frac{\mathrm{2}^{{x}} }{\mathrm{2}^{\left[{x}\right]} }\:{dx}\:= \\ $$

Question Number 26770 Answers: 0 Comments: 1

lim_(n→∞) (((n !)^(1/n) )/n) =

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left({n}\:!\right)^{\mathrm{1}/{n}} }{{n}}\:= \\ $$

Question Number 26768 Answers: 0 Comments: 0

find the nature of U_n =Σ_(p=0) ^(p=n) (1/C_n ^p ) .

$${find}\:{the}\:{nature}\:{of}\:{U}_{{n}} \:\:=\sum_{{p}=\mathrm{0}} ^{{p}={n}} \:\:\frac{\mathrm{1}}{{C}_{{n}} ^{{p}} }\:\:. \\ $$

Question Number 26769 Answers: 1 Comments: 1

∫_( 0) ^∞ (dx/([x+(√(x^2 +1)) ]^3 )) =

$$\underset{\:\mathrm{0}} {\overset{\infty} {\int}}\:\:\frac{{dx}}{\left[{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:\right]^{\mathrm{3}} }\:=\: \\ $$

Question Number 26764 Answers: 0 Comments: 3

find lim_(n−>∝) (1/n) ln( Π_(k=1) ^(n−1) (1− (k/n))).

$${find}\:\:{lim}_{{n}−>\propto} \:\frac{\mathrm{1}}{{n}}\:{ln}\left(\:\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\left(\mathrm{1}−\:\frac{{k}}{{n}}\right)\right). \\ $$

Question Number 26761 Answers: 1 Comments: 0

let put s_n = ((Σ_(k=0) ^(k=n) (2k+1))/(Σ_(k=1) ^(k=n) k)) find lim_(n−>∝) s_n

$${let}\:{put}\:\:{s}_{{n}} =\:\frac{\sum_{{k}=\mathrm{0}} ^{{k}={n}} \left(\mathrm{2}{k}+\mathrm{1}\right)}{\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:{k}} \\ $$$${find}\:{lim}_{{n}−>\propto} {s}_{{n}} \\ $$

Question Number 26759 Answers: 1 Comments: 1

find the value of ∫_0 ^( ∝) (dx/((x+1)(x+2)(x+3))) .

$${find}\:\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\:\propto} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)}\:. \\ $$

Question Number 26758 Answers: 0 Comments: 1

let give D={( x,y )∈R^2 /x^2 −x +y^2 ≤ 4 and 0≤y≤1} calculate ∫∫_D ln(xy)(√( x^2 +y^2 dxdy ))

$${let}\:{give}\:{D}=\left\{\left(\:\:{x},{y}\:\right)\in\mathbb{R}^{\mathrm{2}} /{x}^{\mathrm{2}} −{x}\:+{y}^{\mathrm{2}} \leqslant\:\mathrm{4}\:{and}\:\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\right\} \\ $$$${calculate}\:\int\int_{{D}} {ln}\left({xy}\right)\sqrt{\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} {dxdy}\:\:} \\ $$

Question Number 26757 Answers: 0 Comments: 3

give the decomposition of F(x) = (1/(x^(2n) +1)) inside C[x] then find the value of ∫_0 ^∞ (dx/(1+x^(2n) )) n∈N and n≠o

$${give}\:{the}\:{decomposition}\:{of}\:{F}\left({x}\right)\:=\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}{n}} +\mathrm{1}}\:\:{inside}\:\mathbb{C}\left[{x}\right] \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}{n}} }\:\:\:\:\:\:{n}\in\mathbb{N}\:\:{and}\:{n}\neq{o} \\ $$

Question Number 26756 Answers: 0 Comments: 2

prove that ∫_0 ^1 (dx/(x+ e^x )) = Σ_(n=0) ^∝ (((−1)^n )/((n+1)^(n+1) )) A_n with A_n = ∫_0 ^(n+1) t^n e^(−t) dt .

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{{x}+\:{e}^{{x}} }\:=\:\sum_{{n}=\mathrm{0}} ^{\propto} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+\mathrm{1}\right)^{{n}+\mathrm{1}} }\:{A}_{{n}} \\ $$$${with}\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{{n}+\mathrm{1}} \:{t}^{{n}} \:{e}^{−{t}} {dt}\:. \\ $$

Question Number 26755 Answers: 1 Comments: 0

find ∫ (dx/(x^6 −1)) .

$${find}\:\:\int\:\:\frac{{dx}}{{x}^{\mathrm{6}} −\mathrm{1}}\:\:. \\ $$

Question Number 26751 Answers: 0 Comments: 1

by using fourier serie find the value of Σ_(n=0) ^∝ (1/((2n+1)^2 ))

$${by}\:{using}\:{fourier}\:{serie}\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\propto} \:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 26749 Answers: 0 Comments: 1

let give S_n = Σ_(1≤i<j≤n) (1/(i^2 j^2 )) find lim_(n−>∝) S_n .

$${let}\:{give}\:\:{S}_{{n}} \:=\:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\frac{\mathrm{1}}{{i}^{\mathrm{2}} {j}^{\mathrm{2}} }\:\:\:{find}\:{lim}_{{n}−>\propto} \:\:{S}_{{n}} \:\:. \\ $$

Question Number 26738 Answers: 1 Comments: 1

Question Number 26733 Answers: 1 Comments: 1

Question Number 26732 Answers: 0 Comments: 1

Question Number 27002 Answers: 2 Comments: 4

Question Number 26723 Answers: 0 Comments: 2

Take a point on a given circle as a center and draw an arc which divide the given circle into two equal(in area) regions.Use only Eucledean tools.

$$\mathcal{T}{ake}\:{a}\:{point}\:\boldsymbol{{on}}\:{a}\:{given}\:{circle} \\ $$$${as}\:{a}\:{center}\:{and}\:{draw}\:{an}\:{arc} \\ $$$${which}\:{divide}\:{the}\:{given}\:{circle} \\ $$$${into}\:{two}\:{equal}\left(\mathrm{in}\:\mathrm{area}\right)\:{regions}.{Use}\:{only} \\ $$$${Eucledean}\:{tools}. \\ $$

Question Number 26722 Answers: 0 Comments: 0

lim_(x→−1^+ ) (1+x+x^2 +x^3 +...up to ∞)=? lim_(x→−1^− ) (1+x+x^2 +x^3 +...up to ∞)=?

$$\underset{{x}\rightarrow−\mathrm{1}^{+} } {\mathrm{lim}}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +...\mathrm{up}\:\mathrm{to}\:\infty\right)=? \\ $$$$\underset{{x}\rightarrow−\mathrm{1}^{−} } {\mathrm{lim}}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +...\mathrm{up}\:\mathrm{to}\:\infty\right)=? \\ $$

Question Number 26721 Answers: 0 Comments: 1

−2y(y−12)(y−1)or(y−12)(−2y^2 −2y) both are same.

$$−\mathrm{2}{y}\left({y}−\mathrm{12}\right)\left({y}−\mathrm{1}\right){or}\left({y}−\mathrm{12}\right)\left(−\mathrm{2}{y}^{\mathrm{2}} −\mathrm{2}{y}\right)\:\:{both}\:{are}\:{same}. \\ $$

Question Number 26713 Answers: 0 Comments: 0

2sin (3x)<1

$$\mathrm{2sin}\:\left(\mathrm{3}{x}\right)<\mathrm{1} \\ $$

Question Number 26712 Answers: 1 Comments: 0

If ∣x∣ < 1, then the coefficient of x^n in the expansion of (1+x+x^2 +x^3 +x^4 +...)^2 is

$$\mathrm{If}\:\:\mid{x}\mid\:<\:\mathrm{1},\:\mathrm{then}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{n}} \:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +{x}^{\mathrm{4}} +...\right)^{\mathrm{2}} \:\mathrm{is} \\ $$

Question Number 26711 Answers: 0 Comments: 1

ABC is a right tringle.prove it?

$$\mathrm{ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{right}\:\mathrm{tringle}.\mathrm{prove}\:\mathrm{it}? \\ $$

Question Number 26703 Answers: 0 Comments: 0

The total number of dissimilar terms in the expansion of (x_1 +x_2 +...+x_n )^3 is

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{dissimilar}\:\mathrm{terms}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} +...+{x}_{{n}} \right)^{\mathrm{3}} \:\mathrm{is} \\ $$

Question Number 26694 Answers: 1 Comments: 0

divide x^6 −y^6 by the product of x^2 +x^ y+y^(2 ) and x−y.

$${divide}\:{x}^{\mathrm{6}} −{y}^{\mathrm{6}} \:{by}\:{the}\:{product}\:{of}\:{x}^{\mathrm{2}} +{x}^{} {y}+{y}^{\mathrm{2}\:} \:{and}\:{x}−{y}. \\ $$

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