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Question Number 85198    Answers: 1   Comments: 0

Find x if ^3 (√(x ))+(√x)=(√(12))

$$\mathrm{Find}\:\mathrm{x}\:\mathrm{if}\:\:\:\:^{\mathrm{3}} \sqrt{\mathrm{x}\:}+\sqrt{\mathrm{x}}=\sqrt{\mathrm{12}} \\ $$

Question Number 85195    Answers: 2   Comments: 0

1)find the area between y^2 =3x and y=x^2 −2x 2)∫_0 ^(π/2) ((sin^2 (θ) cos^2 (θ))/((cos^3 (θ)+sin^3 (θ))^2 )) dθ

$$\left.\mathrm{1}\right){find}\:{the}\:{area}\:{between} \\ $$$${y}^{\mathrm{2}} =\mathrm{3}{x}\:{and}\:{y}={x}^{\mathrm{2}} −\mathrm{2}{x} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sin}^{\mathrm{2}} \left(\theta\right)\:{cos}^{\mathrm{2}} \left(\theta\right)}{\left({cos}^{\mathrm{3}} \left(\theta\right)+{sin}^{\mathrm{3}} \left(\theta\right)\right)^{\mathrm{2}} }\:{d}\theta \\ $$

Question Number 85191    Answers: 2   Comments: 0

∫((z+2)/z)

$$\int\frac{\mathrm{z}+\mathrm{2}}{\mathrm{z}} \\ $$

Question Number 85187    Answers: 0   Comments: 0

Show that ∀ n, p ∈ N^(∗ ) C_(n−1) ^( p−1) +C_(n−1) ^( p) =C_n ^( p)

$${Show}\:{that}\:\forall\:{n},\:{p}\:\in\:\mathbb{N}^{\ast\:} \\ $$$${C}_{{n}−\mathrm{1}} ^{\:{p}−\mathrm{1}} +{C}_{{n}−\mathrm{1}} ^{\:{p}} ={C}_{{n}} ^{\:{p}} \\ $$

Question Number 85184    Answers: 0   Comments: 2

Serlea Shows that the sum of the digit of 100^(25) −25 is divisible by 4.

$$\mathrm{Serlea} \\ $$$$\mathrm{Shows}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digit}\:\mathrm{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{100}^{\mathrm{25}} −\mathrm{25} \\ $$$$\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{4}. \\ $$

Question Number 85183    Answers: 0   Comments: 1

Hi veterans Serlea (1) Find the last three digits of: 3005^(11) +3005^(12) +3005^(13) +...+3005^(3002)

$$\mathrm{Hi}\:\mathrm{veterans} \\ $$$$\mathrm{Serlea}\:\left(\mathrm{1}\right) \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{three}\:\mathrm{digits}\:\mathrm{of}: \\ $$$$\mathrm{3005}^{\mathrm{11}} +\mathrm{3005}^{\mathrm{12}} +\mathrm{3005}^{\mathrm{13}} +...+\mathrm{3005}^{\mathrm{3002}} \\ $$$$ \\ $$$$ \\ $$

Question Number 85176    Answers: 2   Comments: 0

what is range function y = (√(x−1)) + (√(5−x))

$$\mathrm{what}\:\mathrm{is}\:\mathrm{range}\: \\ $$$$\mathrm{function}\:\mathrm{y}\:=\:\sqrt{\mathrm{x}−\mathrm{1}}\:+\:\sqrt{\mathrm{5}−\mathrm{x}} \\ $$

Question Number 85169    Answers: 0   Comments: 5

find the n^(th) derivative of function y = (√(sin x)) by Leibniz theorem

$$\mathrm{find}\:\mathrm{the}\:\mathrm{n}^{\mathrm{th}} \:\mathrm{derivative}\:\mathrm{of}\:\mathrm{function} \\ $$$$\mathrm{y}\:=\:\sqrt{\mathrm{sin}\:\mathrm{x}}\:\mathrm{by}\:\mathrm{Leibniz}\:\mathrm{theorem} \\ $$

Question Number 85167    Answers: 0   Comments: 0

let ϕ(x)=Γ(x).Γ(1−x) find ∫_(1/3) ^(1/2) ln(ϕ(x))dx

$${let}\:\varphi\left({x}\right)=\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:\:{find}\:\int_{\frac{\mathrm{1}}{\mathrm{3}}} ^{\frac{\mathrm{1}}{\mathrm{2}}} {ln}\left(\varphi\left({x}\right)\right){dx} \\ $$

Question Number 85166    Answers: 1   Comments: 3

find ∫ (x^2 −1)(√(x^2 +1))dx

$${find}\:\int\:\:\left({x}^{\mathrm{2}} −\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$

Question Number 85165    Answers: 0   Comments: 1

sove (sin^2 x) y^′ +(cosx)y =x

$${sove}\:\:\left({sin}^{\mathrm{2}} {x}\right)\:{y}^{'} \:\:+\left({cosx}\right){y}\:={x} \\ $$

Question Number 85164    Answers: 0   Comments: 3

calculate Σ_(n=0) ^∞ arctan((1/(n^2 +n+1)))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{arctan}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+{n}+\mathrm{1}}\right) \\ $$

Question Number 85163    Answers: 0   Comments: 0

calculate Σ_(n=1) ^∞ arctan((1/(n^2 +n)))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{arctan}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+{n}}\right) \\ $$

Question Number 85162    Answers: 0   Comments: 1

1)find ∫ ln((√x)+(√(x+1)))dx 2) calculate ∫_0 ^1 ln((√x)+(√(x+1)))dx

$$\left.\mathrm{1}\right){find}\:\int\:{ln}\left(\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$

Question Number 85160    Answers: 1   Comments: 2

1) find f(a) =∫_0 ^∞ (dx/(x^4 +a)) with a>0 2) find g(a)=∫_0 ^∞ (dx/((x^4 +a)^2 )) 3) find value of integrals ∫_0 ^∞ (dx/(x^4 +1)) ,∫_0 ^∞ (dx/(2x^4 +8)) ∫_0 ^∞ (dx/((x^4 +1)^2 )) and ∫_0 ^∞ (dx/((2x^4 +8)^2 ))

$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+{a}}\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{g}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{value}\:{of}\:{integrals}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}\:,\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\mathrm{2}{x}^{\mathrm{4}} \:+\mathrm{8}} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{2}} }\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left(\mathrm{2}{x}^{\mathrm{4}} +\mathrm{8}\right)^{\mathrm{2}} } \\ $$

Question Number 85158    Answers: 0   Comments: 0

calculate U_n = ∫_(−(1/n)) ^(1/n) x^2 (√((1−x)/(1+x)))dx (n integr and n≥2) 2) find nature of Σ U_n

$${calculate}\:{U}_{{n}} =\:\int_{−\frac{\mathrm{1}}{{n}}} ^{\frac{\mathrm{1}}{{n}}} \:{x}^{\mathrm{2}} \sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}{dx}\:\:\:\left({n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 85153    Answers: 1   Comments: 1

∫(√(4x^2 −4))dx = ...

$$\int\sqrt{\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}}\boldsymbol{\mathrm{dx}}\:=\:... \\ $$$$ \\ $$$$ \\ $$

Question Number 85148    Answers: 0   Comments: 0

∫_0 ^1 ((1+x^4 )/(1+x^3 +x^7 )) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}+{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{3}} +{x}^{\mathrm{7}} }\:{dx} \\ $$

Question Number 85146    Answers: 2   Comments: 1

find minimum & maximum value of function f(x)= −sin^2 x+sin x−(1/2) , −π≤x≤π

$$\mathrm{find}\:\mathrm{minimum}\:\&\:\mathrm{maximum}\:\mathrm{value}\: \\ $$$$\mathrm{of}\:\mathrm{function}\: \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\:−\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{sin}\:\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\:,\:−\pi\leqslant\mathrm{x}\leqslant\pi \\ $$

Question Number 85142    Answers: 1   Comments: 0

show that ∫_0 ^n [x^2 ]dx =n(n^2 −1)−Σ_(k=1) ^(n^2 −1) (√k)

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{{n}} \left[{x}^{\mathrm{2}} \right]{dx}\:={n}\left({n}^{\mathrm{2}} −\mathrm{1}\right)−\underset{{k}=\mathrm{1}} {\overset{{n}^{\mathrm{2}} −\mathrm{1}} {\sum}}\sqrt{{k}}\: \\ $$

Question Number 85131    Answers: 0   Comments: 4

what procedure will you use to find the inverse of A = ((2,1,9),(1,5,1),(3,0,3) )

$$\mathrm{what}\:\mathrm{procedure}\:\mathrm{will}\:\mathrm{you}\:\mathrm{use}\:\mathrm{to}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{inverse}\:\mathrm{of} \\ $$$$\:\mathrm{A}\:=\:\begin{pmatrix}{\mathrm{2}}&{\mathrm{1}}&{\mathrm{9}}\\{\mathrm{1}}&{\mathrm{5}}&{\mathrm{1}}\\{\mathrm{3}}&{\mathrm{0}}&{\mathrm{3}}\end{pmatrix} \\ $$

Question Number 85130    Answers: 2   Comments: 0

given f(x)= ((√2)+1)sin x +((√2)−1)cos x find masimum value of function [f(x)]^2

$$\mathrm{given}\:\mathrm{f}\left(\mathrm{x}\right)=\:\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)\mathrm{sin}\:\mathrm{x}\:+\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)\mathrm{cos}\:\mathrm{x} \\ $$$$\mathrm{find}\:\mathrm{masimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{function} \\ $$$$\left[\mathrm{f}\left(\mathrm{x}\right)\right]^{\mathrm{2}} \\ $$

Question Number 85129    Answers: 0   Comments: 2

lim_(x→e) [∫_0 ^e ((1/x))dx] =?

$$\underset{{x}\rightarrow{e}} {\mathrm{lim}}\:\left[\underset{\mathrm{0}} {\overset{{e}} {\int}}\left(\frac{\mathrm{1}}{{x}}\right){dx}\right]\:=? \\ $$

Question Number 85127    Answers: 1   Comments: 4

evaluate: lim_(x→0) (√x) ln(sin x)

$$\mathrm{evaluate}: \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt{{x}}\:\mathrm{ln}\left(\mathrm{sin}\:{x}\right) \\ $$$$ \\ $$

Question Number 85116    Answers: 0   Comments: 1

Reduce the equations to Clairaut′s form and find the general solution : x^2 p^2 +yp(2x+y)+y^2 =0 (put y=u and xy=v)

$$\:\mathrm{Reduce}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{to}\:\mathrm{Clairaut}'\mathrm{s}\:\mathrm{form} \\ $$$$\:\mathrm{and}\:\mathrm{find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:: \\ $$$$\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{p}}^{\mathrm{2}} +\boldsymbol{\mathrm{yp}}\left(\mathrm{2}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)+\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{0}\:\:\:\:\:\:\left({put}\:\boldsymbol{{y}}=\boldsymbol{{u}}\:{and}\:\boldsymbol{{xy}}=\boldsymbol{{v}}\right) \\ $$$$\: \\ $$

Question Number 85111    Answers: 1   Comments: 4

Solve the differential equation: ★.(1+x+xy^2 )dy+(y+y^3 )dx

$$\:\boldsymbol{\mathrm{Solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{equation}}: \\ $$$$\:\bigstar.\left(\mathrm{1}+\mathrm{x}+\mathrm{xy}^{\mathrm{2}} \right)\mathrm{dy}+\left(\mathrm{y}+\mathrm{y}^{\mathrm{3}} \right)\mathrm{dx} \\ $$$$\: \\ $$

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