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Question Number 53600 Answers: 0 Comments: 1

calculate A_m =∫_0 ^∞ ((sin(mx))/(e^(2πx) −1)) dx with m>0

$${calculate}\:{A}_{{m}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left({mx}\right)}{{e}^{\mathrm{2}\pi{x}} −\mathrm{1}}\:{dx}\:\:{with}\:{m}>\mathrm{0} \\ $$

Question Number 53599 Answers: 0 Comments: 1

1) calculate A_n =∫_0 ^∞ (x^(n−1) /(e^x +1)) dx with n integr natural (n≥2) 2) find the value of ∫_0 ^∞ (x/(e^x +1))dx

$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{{n}−\mathrm{1}} }{{e}^{{x}} \:+\mathrm{1}}\:{dx}\:\:\:{with}\:{n}\:{integr}\:{natural}\:\:\left({n}\geqslant\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}}{{e}^{{x}} \:+\mathrm{1}}{dx} \\ $$

Question Number 53618 Answers: 1 Comments: 3

Let x,y∈Z if x^2 +y^2 divide xy+1 then prove ((x^2 +y^2 )/(xy+1)) is square of integer number

$$\mathrm{Let}\:\mathrm{x},\mathrm{y}\in\mathbb{Z} \\ $$$$\mathrm{if}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \:\mathrm{divide}\:\mathrm{xy}+\mathrm{1}\:\mathrm{then}\:\mathrm{prove} \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }{\mathrm{xy}+\mathrm{1}}\:\mathrm{is}\:\mathrm{square}\:\mathrm{of}\:\mathrm{integer}\:\mathrm{number} \\ $$

Question Number 53595 Answers: 1 Comments: 5

Question Number 53593 Answers: 2 Comments: 1

Question Number 53570 Answers: 1 Comments: 7

Question Number 53556 Answers: 2 Comments: 1

Find the value of k for which the system of linear equations kx+2y=5 and 3x+y=1 has zero solutions.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:{k}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the} \\ $$$$\mathrm{system}\:\mathrm{of}\:\mathrm{linear}\:\mathrm{equations}\:\:{kx}+\mathrm{2}{y}=\mathrm{5}\: \\ $$$$\mathrm{and}\:\:\:\mathrm{3}{x}+{y}=\mathrm{1}\:\mathrm{has}\:\mathrm{zero}\:\mathrm{solutions}. \\ $$

Question Number 53550 Answers: 1 Comments: 2

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

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

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