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Question Number 53532    Answers: 1   Comments: 1

The derivative of F (x)=∫_x^2 ^x^3 (1/(log t)) dt (x >0) is

$$\mathrm{The}\:\mathrm{derivative}\:\mathrm{of}\:{F}\:\left({x}\right)=\underset{{x}^{\mathrm{2}} } {\overset{{x}^{\mathrm{3}} } {\int}}\:\frac{\mathrm{1}}{\mathrm{log}\:{t}}\:{dt} \\ $$$$\left({x}\:>\mathrm{0}\right)\:\mathrm{is} \\ $$

Question Number 53528    Answers: 0   Comments: 0

x^3 + y^3 + z^3 = x + y + z x^2 + y^2 + z^2 = xyz x, y, z ∈ R^+ How many triple of (x, y, z) ?

$${x}^{\mathrm{3}} \:+\:{y}^{\mathrm{3}} \:+\:{z}^{\mathrm{3}} \:\:=\:\:{x}\:+\:{y}\:+\:{z} \\ $$$${x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:\:=\:\:{xyz} \\ $$$${x},\:{y},\:{z}\:\:\in\:\mathbb{R}^{+} \\ $$$${How}\:\:{many}\:\:{triple}\:\:{of}\:\:\left({x},\:{y},\:{z}\right)\:\:? \\ $$

Question Number 53530    Answers: 2   Comments: 1

Question Number 53522    Answers: 1   Comments: 0

Cost of 2 pencils and 3 erasers is Rs.18, while the cost of 1 pencil and 2 erasers is Rs. 11. Find the cost of each pencil.

$$\mathrm{Cost}\:\mathrm{of}\:\:\mathrm{2}\:\mathrm{pencils}\:\mathrm{and}\:\:\mathrm{3}\:\mathrm{erasers}\:\mathrm{is}\:\mathrm{Rs}.\mathrm{18}, \\ $$$$\mathrm{while}\:\mathrm{the}\:\mathrm{cost}\:\mathrm{of}\:\:\mathrm{1}\:\:\mathrm{pencil}\:\mathrm{and}\:\:\mathrm{2}\:\mathrm{erasers} \\ $$$$\mathrm{is}\:\mathrm{Rs}.\:\mathrm{11}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{cost}\:\mathrm{of}\:\mathrm{each}\:\mathrm{pencil}. \\ $$

Question Number 53518    Answers: 2   Comments: 0

Find the sum of 3.2^− and 5.4^− .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\:\mathrm{3}.\overset{−} {\mathrm{2}}\:\:\mathrm{and}\:\:\mathrm{5}.\overset{−} {\mathrm{4}}\:\:. \\ $$

Question Number 53515    Answers: 1   Comments: 0

What is the sum of ten odd numbers and eleven even numbers?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{ten}\:\mathrm{odd}\:\mathrm{numbers} \\ $$$$\mathrm{and}\:\mathrm{eleven}\:\mathrm{even}\:\mathrm{numbers}? \\ $$

Question Number 53514    Answers: 0   Comments: 0

What is the sum of ten odd numbers and eleven even numbers?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{ten}\:\mathrm{odd}\:\mathrm{numbers} \\ $$$$\mathrm{and}\:\mathrm{eleven}\:\mathrm{even}\:\mathrm{numbers}? \\ $$

Question Number 53513    Answers: 0   Comments: 0

What is the sum of ten odd numbers and eleven even numbers?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{ten}\:\mathrm{odd}\:\mathrm{numbers} \\ $$$$\mathrm{and}\:\mathrm{eleven}\:\mathrm{even}\:\mathrm{numbers}? \\ $$

Question Number 53502    Answers: 2   Comments: 0

If (x−5) is a factor of 2x^2 +2px−2p=0, the value of p is

$$\mathrm{If}\:\left({x}−\mathrm{5}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\:\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{px}−\mathrm{2}{p}=\mathrm{0}, \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:{p}\:\:\mathrm{is} \\ $$

Question Number 53501    Answers: 0   Comments: 0

If (x−5) is a factor of 2x^2 +2px−2p=0, the value of p is

$$\mathrm{If}\:\left({x}−\mathrm{5}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of}\:\:\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{px}−\mathrm{2}{p}=\mathrm{0}, \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:{p}\:\:\mathrm{is} \\ $$

Question Number 53500    Answers: 2   Comments: 0

1. If (√((x+2)^(2 ) +y^2 ))+(√((x−2)^2 +y^2 ))=6 show that when the equation is simplified, it can be express as (x^2 /9)+(y^2 /5)=1 2. find the value of n such that the linear factors of the form x+ay+b and x+cy+d with integer coefficients have the product x^2 +5xy+x+ny−n sir please help

$$\mathrm{1}.\:{If}\:\:\sqrt{\left({x}+\mathrm{2}\right)^{\mathrm{2}\:} +{y}^{\mathrm{2}} }+\sqrt{\left({x}−\mathrm{2}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} }=\mathrm{6} \\ $$$${show}\:{that}\:{when}\:{the}\:{equation}\:{is}\: \\ $$$${simplified},\:{it}\:{can}\:{be}\:{express}\:{as} \\ $$$$\frac{{x}^{\mathrm{2}} }{\mathrm{9}}+\frac{{y}^{\mathrm{2}} }{\mathrm{5}}=\mathrm{1} \\ $$$$\mathrm{2}.\:{find}\:{the}\:{value}\:{of}\:{n}\:{such}\:{that}\:{the} \\ $$$${linear}\:{factors}\:{of}\:{the}\:{form}\:{x}+{ay}+{b}\: \\ $$$${and}\:{x}+{cy}+{d}\:{with}\:{integer}\:{coefficients} \\ $$$${have}\:{the}\:{product}\:{x}^{\mathrm{2}} +\mathrm{5}{xy}+{x}+{ny}−{n} \\ $$$${sir}\:{please}\:{help} \\ $$

Question Number 53492    Answers: 1   Comments: 1

Question Number 53491    Answers: 3   Comments: 0

(((1+x)/(√x)))^2 +2a(((1+x)/(√x)))+1=0 solve for x.

$$\left(\frac{\mathrm{1}+{x}}{\sqrt{{x}}}\right)^{\mathrm{2}} +\mathrm{2}{a}\left(\frac{\mathrm{1}+{x}}{\sqrt{{x}}}\right)+\mathrm{1}=\mathrm{0} \\ $$$${solve}\:{for}\:{x}. \\ $$

Question Number 53483    Answers: 0   Comments: 9

Question Number 53477    Answers: 1   Comments: 1

let f(a)=∫_0 ^1 (dt/((√(x+a)) +3)) 1) calculate f(a) 2) find also ∫_0 ^1 (dt/((√(x+a))((√(x+a)) +3)^2 )) 3) find the values of integrals ∫_0 ^1 (dt/((√(x+1))+3)) and ∫_0 ^1 (dt/((√(x+1))((√(x+1))+3)^2 ))

$${let}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{{x}+{a}}\:+\mathrm{3}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{{x}+{a}}\left(\sqrt{{x}+{a}}\:+\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{{x}+\mathrm{1}}+\mathrm{3}}\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\sqrt{{x}+\mathrm{1}}\left(\sqrt{{x}+\mathrm{1}}+\mathrm{3}\right)^{\mathrm{2}} } \\ $$

Question Number 53476    Answers: 0   Comments: 0

let f(x)=∫_0 ^x t(√(2t−1))dt calculate ∣sup_(1≤x≤2) f(x) −inf_(1≤x≤2) f(x)∣

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{t}\sqrt{\mathrm{2}{t}−\mathrm{1}}{dt}\:\:\:\:{calculate}\:\mid{sup}_{\mathrm{1}\leqslant{x}\leqslant\mathrm{2}} \:{f}\left({x}\right)\:−{inf}_{\mathrm{1}\leqslant{x}\leqslant\mathrm{2}} {f}\left({x}\right)\mid \\ $$

Question Number 53474    Answers: 1   Comments: 0

calculate ∫_0 ^1 ((5^(2x+1) −2^(2x−1) )/(10^x )) dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{5}^{\mathrm{2}{x}+\mathrm{1}} \:−\mathrm{2}^{\mathrm{2}{x}−\mathrm{1}} }{\mathrm{10}^{{x}} }\:{dx} \\ $$

Question Number 53472    Answers: 1   Comments: 2

Question Number 53471    Answers: 1   Comments: 3

1)find U_n = ∫_0 ^(π/4) tan^n tdt with n integr . 2) find lim_(n→+∞) U_n 3) calculate Σ_(n=0) ^∞ U_n

$$\left.\mathrm{1}\right){find}\:\:{U}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{tan}^{{n}} {tdt}\:\:\:{with}\:{n}\:{integr}\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{U}_{{n}} \\ $$$$ \\ $$

Question Number 53470    Answers: 0   Comments: 1

find Vn=∫_(1/n) ^((an−1)/n) ((√x)/(√(a−(√x)+x)))dx

$${find}\:\:{Vn}=\int_{\frac{\mathrm{1}}{{n}}} ^{\frac{{an}−\mathrm{1}}{{n}}} \:\frac{\sqrt{{x}}}{\sqrt{{a}−\sqrt{{x}}+{x}}}{dx} \\ $$$$ \\ $$

Question Number 53468    Answers: 1   Comments: 3

Question Number 53467    Answers: 1   Comments: 0

let A_(n m) =∫_0 ^1 x^n (1−x)^m dx with n and n integrs naturals 1) calculate A_(n m) by using factoriels 2) find Σ_(n,m) A_(nm)

$${let}\:{A}_{{n}\:{m}} \:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \left(\mathrm{1}−{x}\right)^{{m}} {dx}\:\:{with}\:{n}\:{and}\:{n}\:{integrs}\:{naturals} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}\:{m}} \:\:{by}\:{using}\:{factoriels} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n},{m}} \:{A}_{{nm}} \\ $$

Question Number 53466    Answers: 1   Comments: 0

let U_n =(1/n){Π_(k=1) ^n (n+k)}^(1/n) find lim_(n→+∞) U_n

$${let}\:{U}_{{n}} =\frac{\mathrm{1}}{{n}}\left\{\prod_{{k}=\mathrm{1}} ^{{n}} \left({n}+{k}\right)\right\}^{\frac{\mathrm{1}}{{n}}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$

Question Number 53465    Answers: 1   Comments: 1

find ∫_(−(π/2)) ^(π/2) (√(cosx −cos^3 x))dx

$${find}\:\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{cosx}\:−{cos}^{\mathrm{3}} {x}}{dx} \\ $$

Question Number 53464    Answers: 1   Comments: 1

let U_n = (((∫_0 ^n e^(−x^2 ) dx)^2 )/(∫_0 ^n e^(−nx^2 ) dx)) 1) calculate lim_(n→+∞) U_n 2) determne nature of Σ U_n and Σ U_n ^3 .

$${let}\:{U}_{{n}} =\:\frac{\left(\int_{\mathrm{0}} ^{{n}} \:{e}^{−{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} }{\int_{\mathrm{0}} ^{{n}} \:\:{e}^{−{nx}^{\mathrm{2}} } {dx}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{determne}\:{nature}\:{of}\:\Sigma\:\:{U}_{{n}} \:\:{and}\:\Sigma\:{U}_{{n}} ^{\mathrm{3}} \:. \\ $$

Question Number 53463    Answers: 1   Comments: 1

1)let 0<θ<(π/2) and A(θ) =∫_0 ^(π/2) (dx/(√(x^2 +2sinθ x +1))) calculate A(θ) 2) calculate ∫_0 ^(π/2) (dx/(√(x^2 +(√2)x +1)))

$$\left.\mathrm{1}\right){let}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}}\:\:\:\:{and}\:\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{sin}\theta\:{x}\:+\mathrm{1}}} \\ $$$${calculate}\:{A}\left(\theta\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+\sqrt{\mathrm{2}}{x}\:+\mathrm{1}}} \\ $$

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