Question and Answers Forum

AllQuestion and Answers: Page 1147

Question Number 102510 Answers: 2 Comments: 0

lim_(△x→0) ((sin^2 ((1/3)x+(1/3)△x)−sin^2 (1/3)x)/(△x))=?

$${li}\underset{\bigtriangleup{x}\rightarrow\mathrm{0}} {{m}}\frac{{sin}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{3}}{x}+\frac{\mathrm{1}}{\mathrm{3}}\bigtriangleup{x}\right)−{sin}^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{3}}{x}}{\bigtriangleup{x}}=? \\ $$

Question Number 102508 Answers: 0 Comments: 5

lim_(△x→0) ((e^(sin(x−△x)) −e^(sinx) )/(△x))=?

$${li}\underset{\bigtriangleup{x}\rightarrow\mathrm{0}} {{m}}\frac{{e}^{{sin}\left({x}−\bigtriangleup{x}\right)} −{e}^{{sinx}} }{\bigtriangleup{x}}=? \\ $$

Question Number 102507 Answers: 0 Comments: 2

lim_(x→e^(lnx) ) ((x/e^(lnx) ))^(−logx) =?

$${li}\underset{{x}\rightarrow{e}^{{lnx}} } {{m}}\left(\frac{{x}}{{e}^{{lnx}} }\right)^{−{logx}} =? \\ $$

Question Number 102491 Answers: 0 Comments: 0

Lets p ∈N and n∈N^∗ A_n =2^n +p and d_n =PGCD(A_n ,A_(n+1) ) 1) show that d_n /2^n 2)determine the parity of A_n as a function of that of p 3)determine the parity of d_n as a function of that of p 4)deduce pgcd(2^(2009) +2009,2^(2010) +2009)

$$ \\ $$$$\boldsymbol{{L}}{e}\boldsymbol{{ts}}\:\boldsymbol{{p}}\:\in\mathbb{N}\:\boldsymbol{{and}}\:\boldsymbol{{n}}\in\mathbb{N}^{\ast} \\ $$$$\boldsymbol{{A}}_{{n}} =\mathrm{2}^{\boldsymbol{{n}}} +\boldsymbol{{p}}\:\boldsymbol{{and}}\:\boldsymbol{{d}}_{\boldsymbol{{n}}} =\boldsymbol{{P}}{GCD}\left(\boldsymbol{{A}}_{{n}} ,\boldsymbol{{A}}_{\boldsymbol{{n}}+\mathrm{1}} \right) \\ $$$$\left.\mathrm{1}\right)\:\boldsymbol{{show}}\:\boldsymbol{{that}}\:\:\boldsymbol{{d}}_{\boldsymbol{{n}}} /\mathrm{2}^{\boldsymbol{{n}}} \\ $$$$\left.\mathrm{2}\right)\boldsymbol{{determine}}\:\boldsymbol{{the}}\:\boldsymbol{{parity}}\:{of}\:\boldsymbol{{A}}_{{n}} \:{as}\:{a}\:{function}\:{of}\:{that}\:\boldsymbol{{of}}\:\boldsymbol{{p}} \\ $$$$\left.\mathrm{3}\right){determine}\:{the}\:{parity}\:{of}\:\boldsymbol{{d}}_{\boldsymbol{{n}}} \:{as}\:{a}\:{function}\:{of}\:{that}\:{of}\:\boldsymbol{{p}} \\ $$$$\left.\mathrm{4}\right){deduce}\:{pgcd}\left(\mathrm{2}^{\mathrm{2009}} +\mathrm{2009},\mathrm{2}^{\mathrm{2010}} +\mathrm{2009}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 102490 Answers: 1 Comments: 0

x^3 −bx−c=0 ; b, c >0 ; ((b/3))^3 >((c/2))^2 To find the three real roots without the use of trigonometric solution to cubic polynomial...

$${x}^{\mathrm{3}} −{bx}−{c}=\mathrm{0}\:\:\:\:\:;\:\:{b},\:{c}\:>\mathrm{0}\:;\:\:\left(\frac{{b}}{\mathrm{3}}\right)^{\mathrm{3}} >\left(\frac{{c}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${To}\:{find}\:{the}\:{three}\:{real}\:{roots}\:{without} \\ $$$${the}\:{use}\:{of}\:{trigonometric}\:{solution} \\ $$$${to}\:{cubic}\:{polynomial}... \\ $$

Question Number 102484 Answers: 1 Comments: 0

Question Number 102480 Answers: 1 Comments: 0

n positive integer. when dividen by 7 give remainder 4 and when divided by 4 give remainder 2. find the value of n

$${n}\:{positive}\:{integer}.\:{when} \\ $$$${dividen}\:{by}\:\mathrm{7}\:{give}\:{remainder}\:\mathrm{4} \\ $$$${and}\:{when}\:{divided}\:{by}\:\mathrm{4}\:{give} \\ $$$${remainder}\:\mathrm{2}.\:{find}\:{the}\:{value} \\ $$$${of}\:{n}\: \\ $$

Question Number 102474 Answers: 1 Comments: 0

Un=(1+(√2))^n show that we have p_n ∈N / U_n =(√p_n )+(√(p_n +1))

$$\boldsymbol{{U}}{n}=\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)^{{n}} \\ $$$${show}\:{that}\:{we}\:{have}\:\boldsymbol{{p}}_{\boldsymbol{{n}}} \in\mathbb{N}\:/ \\ $$$$\boldsymbol{{U}}_{\boldsymbol{{n}}} =\sqrt{{p}_{{n}} }+\sqrt{{p}_{{n}} +\mathrm{1}} \\ $$

Question Number 102470 Answers: 1 Comments: 0

∫(dx/(x^(10) +x^2 ))

$$\int\frac{{dx}}{{x}^{\mathrm{10}} +{x}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 102467 Answers: 1 Comments: 5

hello friends, is zero an even or odd number or not both?

$${hello}\:{friends},\:{is}\:{zero}\:{an}\: \\ $$$${even}\:{or}\:{odd}\:{number}\:{or}\:{not}\: \\ $$$${both}? \\ $$

Question Number 102462 Answers: 8 Comments: 0

Calculate ; J=∫(dx/(x(x^2 +x−1)^2 )) K=∫((x^3 +x−1)/((x^2 +2)^2 ))dx L=∫(dx/(x+(√(x^2 +1))))

$$\mathrm{Calculate}\:; \\ $$$$\mathrm{J}=\int\frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\mathrm{K}=\int\frac{\mathrm{x}^{\mathrm{3}} +\mathrm{x}−\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$$$\mathrm{L}=\int\frac{\mathrm{dx}}{\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$

Question Number 102461 Answers: 1 Comments: 0

∫ (dx/(√(5e^(2x) +4e^x +1))) =?

$$\int\:\frac{{dx}}{\sqrt{\mathrm{5}{e}^{\mathrm{2}{x}} +\mathrm{4}{e}^{{x}} +\mathrm{1}}}\:=? \\ $$

Question Number 102555 Answers: 0 Comments: 0

∫_0 ^1 ((x^(98) −99x+98)/(logx))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{98}} −\mathrm{99}{x}+\mathrm{98}}{{logx}}{dx} \\ $$

Question Number 102450 Answers: 3 Comments: 2

Question Number 102444 Answers: 1 Comments: 0

find the area bounded the curves y^2 = 36+12x and y^2 =16−8x

$${find}\:{the}\:{area}\:{bounded}\:{the} \\ $$$${curves}\:{y}^{\mathrm{2}} =\:\mathrm{36}+\mathrm{12}{x}\:{and}\: \\ $$$${y}^{\mathrm{2}} =\mathrm{16}−\mathrm{8}{x}\: \\ $$

Question Number 102418 Answers: 1 Comments: 0

Question Number 102417 Answers: 3 Comments: 0

calculate ∫_0 ^∞ e^(−x) ln(1+e^x )dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx} \\ $$

Question Number 102416 Answers: 2 Comments: 0

calculate ∫_0 ^1 e^(−x) ln(1+e^x )dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{e}^{−\mathrm{x}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx} \\ $$

Question Number 102407 Answers: 0 Comments: 0

Sir MJS & Sir John Santu You both have decided to leave this forum for different reasons. Being agree with your reasons and accepting your right of decision I dare to suggest not to disconnect fully from the forum.Please stay connected although for very short time on daily/weekly basis.This is also necessary because we have no means to contact you. After all this is only a request. You may or may not accept it.

$$\mathrm{Sir}\:\mathrm{MJS}\:\&\:\mathrm{Sir}\:\mathrm{John}\:\mathrm{Santu} \\ $$$${You}\:{both}\:{have}\:{decided}\:{to}\:{leave} \\ $$$${this}\:{forum}\:{for}\:{different}\:{reasons}. \\ $$$${Being}\:{agree}\:{with}\:{your}\:{reasons} \\ $$$${and}\:{accepting}\:{your}\:{right}\:{of}\: \\ $$$${decision}\:{I}\:{dare}\:{to}\:{suggest}\:{not} \\ $$$${to}\:{disconnect}\:{fully}\:{from}\:{the} \\ $$$${forum}.{Please}\:{stay}\:{connected} \\ $$$${although}\:{for}\:{very}\:{short}\:{time}\:{on} \\ $$$${daily}/{weekly}\:{basis}.\mathcal{T}{his}\:{is}\:{also} \\ $$$${necessary}\:{because}\:{we}\:{have}\:{no} \\ $$$${means}\:{to}\:{contact}\:{you}. \\ $$$$\:\:\:\:\mathcal{A}{fter}\:{all}\:{this}\:{is}\:{only}\:{a}\:{request}. \\ $$$${You}\:{may}\:{or}\:{may}\:{not}\:{accept}\:{it}. \\ $$$$ \\ $$

Question Number 102397 Answers: 0 Comments: 0

∫ln(x−(√x)+1)dx

$$\int{ln}\left({x}−\sqrt{{x}}+\mathrm{1}\right){dx} \\ $$

Question Number 102383 Answers: 0 Comments: 2

∫(x/(sin^2 x−3))

$$\int\frac{{x}}{{sin}^{\mathrm{2}} {x}−\mathrm{3}} \\ $$

Question Number 102382 Answers: 1 Comments: 0

x^2 ∙(dy/dx)=x^2 +xy+y^2

$$\mathrm{x}^{\mathrm{2}} \centerdot\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x}^{\mathrm{2}} +\mathrm{xy}+\mathrm{y}^{\mathrm{2}} \\ $$

Question Number 102381 Answers: 0 Comments: 0

∫((x(√6)sec^2 (x/2))/(1+9tan^4 (x/2)+18tan^2 (x/2)))dx