Question and Answers Forum

All Questions   Topic List

AllQuestion and Answers: Page 564

Question Number 162177    Answers: 1   Comments: 0

𝛗 = ∫_0 ^( 1) (( ln^( 2) ( x ). Li_( 2) (x ))/x^ ) dx =?

$$ \\ $$$$\:\:\:\boldsymbol{\phi}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{ln}^{\:\mathrm{2}} \left(\:{x}\:\right).\:\mathrm{Li}_{\:\mathrm{2}} \:\left({x}\:\right)}{{x}^{\:} }\:{dx}\:=? \\ $$

Question Number 162174    Answers: 0   Comments: 1

x^( 2) βˆ’ 4x βˆ’1=0 Ξ± , Ξ² are roots Ξ±^( 3) + 17Ξ² +5 =? βˆ’βˆ’βˆ’solutionβˆ’βˆ’βˆ’ Ξ± is root β‡’ Ξ±^( 2) βˆ’4Ξ± βˆ’1=0 β‡’ Ξ±^( 2) = 4Ξ± +1 βœ“ Ξ±^( 3) + 17Ξ² +5 = Ξ± . Ξ±^( 2) +17Ξ² +5 = Ξ± ( 4Ξ± +1 )+ 17Ξ² +5 = 4Ξ±^( 2) + Ξ± + 17Ξ² +5 = 4 (4Ξ± +1 )+Ξ± +17Ξ² +5=17(Ξ±+Ξ²)+9 = 17S +9= 17 (4 )+9=77

$$ \\ $$$$\:\:\:\:{x}^{\:\mathrm{2}} βˆ’\:\mathrm{4}{x}\:βˆ’\mathrm{1}=\mathrm{0}\:\: \\ $$$$\:\:\:\:\:\alpha\:,\:\beta\:\:{are}\:{roots}\: \\ $$$$\:\:\:\:\:\alpha^{\:\mathrm{3}} \:+\:\mathrm{17}\beta\:+\mathrm{5}\:=? \\ $$$$\:\:βˆ’βˆ’βˆ’{solution}βˆ’βˆ’βˆ’ \\ $$$$\:\:\:\alpha\:\:\:{is}\:{root}\:\:\:\Rightarrow\:\alpha^{\:\mathrm{2}} βˆ’\mathrm{4}\alpha\:βˆ’\mathrm{1}=\mathrm{0} \\ $$$$\:\:\:\:\:\Rightarrow\:\alpha^{\:\mathrm{2}} =\:\mathrm{4}\alpha\:+\mathrm{1}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\alpha^{\:\mathrm{3}} +\:\mathrm{17}\beta\:+\mathrm{5}\:=\:\alpha\:.\:\alpha^{\:\mathrm{2}} +\mathrm{17}\beta\:+\mathrm{5} \\ $$$$\:\:=\:\alpha\:\left(\:\mathrm{4}\alpha\:+\mathrm{1}\:\right)+\:\mathrm{17}\beta\:+\mathrm{5} \\ $$$$\:\:=\:\mathrm{4}\alpha^{\:\mathrm{2}} +\:\alpha\:+\:\mathrm{17}\beta\:+\mathrm{5} \\ $$$$\:\:=\:\mathrm{4}\:\left(\mathrm{4}\alpha\:+\mathrm{1}\:\right)+\alpha\:+\mathrm{17}\beta\:+\mathrm{5}=\mathrm{17}\left(\alpha+\beta\right)+\mathrm{9} \\ $$$$\:\:=\:\mathrm{17S}\:+\mathrm{9}=\:\mathrm{17}\:\left(\mathrm{4}\:\right)+\mathrm{9}=\mathrm{77} \\ $$$$ \\ $$

Question Number 162171    Answers: 0   Comments: 0

Question Number 162169    Answers: 2   Comments: 0

determinant ((( determinant ((( x+y+x^2 y^2 =586_(x=?,y=? ) ^(x,y∈Z ) )))_ ^ _() ^(β€’) )))

$$ \\ $$$$\:\:\:\:\:\:\:\begin{array}{|c|}{\overset{\bullet} {\:\:\:\:\:\begin{array}{|c|}{\:\:\:\underset{{x}=?,{y}=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {\overset{{x},{y}\in\mathbb{Z}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {{x}+{y}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} =\mathrm{586}}}\:\:}\\\hline\end{array}_{} ^{} }\:\:\:\:}\\\hline\end{array} \\ $$$$ \\ $$

Question Number 162168    Answers: 0   Comments: 0

Solve the integroβˆ’differential equation: i(t) + 4(di/dt) + ∫i(t)dt = 2 cos (3t+ 60Β°) where i(t) is a sinulsodial current.

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{integro}βˆ’\mathrm{differential} \\ $$$$\mathrm{equation}: \\ $$$$\:{i}\left({t}\right)\:+\:\mathrm{4}\frac{{di}}{{dt}}\:+\:\int{i}\left({t}\right){dt}\:=\:\mathrm{2}\:\mathrm{cos}\:\left(\mathrm{3}{t}+\:\mathrm{60}Β°\right) \\ $$$$\mathrm{where}\:{i}\left({t}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{sinulsodial}\:\mathrm{current}. \\ $$

Question Number 162151    Answers: 1   Comments: 0

Question Number 162116    Answers: 2   Comments: 0

A function f is such that f : R β†’ R where f(xy+1) = f(x)βˆ™f(y)βˆ’f(y)βˆ’x+2 , βˆ€x,y ∈ R . Find value of 10βˆ™f(2017)+f(0) .

$${A}\:{function}\:\:{f}\:\:\:{is}\:\:{such}\:\:{that}\:\:{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{R}\:\:{where} \\ $$$$\:\:\:{f}\left({xy}+\mathrm{1}\right)\:=\:{f}\left({x}\right)\centerdot{f}\left({y}\right)βˆ’{f}\left({y}\right)βˆ’{x}+\mathrm{2}\:\:,\:\:\forall{x},{y}\:\in\:\mathbb{R}\:. \\ $$$${Find}\:\:{value}\:\:{of}\:\:\mathrm{10}\centerdot{f}\left(\mathrm{2017}\right)+{f}\left(\mathrm{0}\right)\:. \\ $$

Question Number 162112    Answers: 1   Comments: 0

∫(( cos(x))/((1βˆ’cos(x))^2 ))dx=?

$$\int\frac{\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}{\left(\mathrm{1}βˆ’\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\right)^{\mathrm{2}} }\boldsymbol{{dx}}=? \\ $$

Question Number 162109    Answers: 1   Comments: 0

Question Number 162103    Answers: 1   Comments: 0

Find: lim_(x→0) ((sinx - sin^(-1) x)/(sinhx - sinh^(-1) x)) = ?

$$\mathrm{Find}: \\ $$$$\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sinx}\:-\:\mathrm{sin}^{-\mathrm{1}} \mathrm{x}}{\mathrm{sinhx}\:-\:\mathrm{sinh}^{-\mathrm{1}} \mathrm{x}}\:=\:? \\ $$

Question Number 162102    Answers: 0   Comments: 2

Solve for real numbers: 5^x + 4^(1/x) + 25^x βˆ™ 16^(1/x) = 2527

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{5}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{4}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} \:+\:\mathrm{25}^{\boldsymbol{\mathrm{x}}} \:\centerdot\:\mathrm{16}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} \:=\:\mathrm{2527} \\ $$

Question Number 162100    Answers: 1   Comments: 1

Question Number 162099    Answers: 0   Comments: 0

PROVE THAT Ξ©= ∫_0 ^( 1) (( Li_( 2) (x ). ln( x ))/x) dx =^? (( βˆ’Ο€^( 4) )/(90)) βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ Ξ©= ∫_0 ^( 1) ln (x )Ξ£_(n=1) ^∞ (( x^( nβˆ’1) )/n^( 2) ) dx = Ξ£_(n=1) ^∞ (1/n^( 2) ) ∫_0 ^( 1) x^( nβˆ’1) . ln(x ) dx = Ξ£_(n=1) ^∞ (1/n^( 2) ) {[ (x^( n) /n) ln( x )]_0 ^( 1) βˆ’(1/n) ∫_0 ^( 1) x^( nβˆ’1) dx} = Ξ£_(n=1) ^∞ ((βˆ’1)/n^( 4) ) = βˆ’ ΞΆ (4 ) = ((βˆ’Ο€^( 4) )/( 90)) β–  m.n βˆ’βˆ’βˆ’ M . N βˆ’βˆ’βˆ’

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathscr{PROVE}\:\:\:\mathscr{THAT}\:\: \\ $$$$\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{Li}_{\:\mathrm{2}} \:\left({x}\:\right).\:\mathrm{ln}\left(\:{x}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\:βˆ’\pi^{\:\mathrm{4}} }{\mathrm{90}} \\ $$$$\:\:\:\:\:βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ \\ $$$$\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\:\left({x}\:\right)\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{x}^{\:{n}βˆ’\mathrm{1}} }{{n}^{\:\mathrm{2}} }\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\:\mathrm{2}} }\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:{x}^{\:{n}βˆ’\mathrm{1}} .\:\mathrm{ln}\left({x}\:\right)\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\:\mathrm{2}} }\:\left\{\left[\:\frac{{x}^{\:{n}} }{{n}}\:\mathrm{ln}\left(\:{x}\:\right)\right]_{\mathrm{0}} ^{\:\mathrm{1}} βˆ’\frac{\mathrm{1}}{{n}}\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\:{n}βˆ’\mathrm{1}} {dx}\right\} \\ $$$$\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{βˆ’\mathrm{1}}{{n}^{\:\mathrm{4}} }\:=\:βˆ’\:\zeta\:\left(\mathrm{4}\:\right)\:=\:\frac{βˆ’\pi^{\:\mathrm{4}} }{\:\mathrm{90}}\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:βˆ’βˆ’βˆ’\:\mathscr{M}\:.\:\mathscr{N}\:\:βˆ’βˆ’βˆ’\: \\ $$$$ \\ $$

Question Number 162139    Answers: 2   Comments: 0

Question Number 162138    Answers: 1   Comments: 0

Question Number 162118    Answers: 0   Comments: 7

determiner le reste de la division euclidienne de 10^(100) par 105

$${determiner}\:{le}\:{reste}\:{de}\:{la}\:{division}\:{euclidienne}\:{de} \\ $$$$\mathrm{10}^{\mathrm{100}} \:{par}\:\mathrm{105} \\ $$

Question Number 162117    Answers: 2   Comments: 1

Question Number 162092    Answers: 0   Comments: 0

Question Number 162088    Answers: 2   Comments: 0

x^2 =2^x solve for x=?

$${x}^{\mathrm{2}} =\mathrm{2}^{{x}} \\ $$$${solve}\:\:\:{for}\:\:\:\:{x}=? \\ $$

Question Number 162083    Answers: 0   Comments: 0

Question Number 162081    Answers: 2   Comments: 0

lim_(xβ†’0) ((√((1/x)+(√((1/x)+(√(1/x)))))) βˆ’(√((1/x)βˆ’(√((1/x)+(√(1/x)))))) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}}}}\:βˆ’\sqrt{\frac{\mathrm{1}}{{x}}βˆ’\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}}}}\:=?\right. \\ $$

Question Number 162165    Answers: 2   Comments: 1

If the ratio of the roots of equation ax^2 +2bx+c=0 is same as the ratio of the roots of px^2 +2qx+r=0 where a,b,c,p ,r are non zero real numbers . Then the value of ((b^2 /q^2 ))((p/a))((r/c)) is equal to

$$\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{equation} \\ $$$$\:{ax}^{\mathrm{2}} +\mathrm{2}{bx}+{c}=\mathrm{0}\:\mathrm{is}\:\mathrm{same}\:\mathrm{as}\:\mathrm{the}\:\mathrm{ratio} \\ $$$$\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{px}^{\mathrm{2}} +\mathrm{2}{qx}+{r}=\mathrm{0}\:\mathrm{where} \\ $$$$\:{a},{b},{c},{p}\:,{r}\:\mathrm{are}\:\mathrm{non}\:\mathrm{zero}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\left(\frac{{b}^{\mathrm{2}} }{{q}^{\mathrm{2}} }\right)\left(\frac{{p}}{{a}}\right)\left(\frac{{r}}{{c}}\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\: \\ $$

Question Number 162074    Answers: 1   Comments: 0

Find the exact value of Ξ£_(k=0) ^(1004) ( _k ^(2014) ) βˆ™ 3^k .

$${Find}\:\:{the}\:\:{exact}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{1004}} {\sum}}\:\left(\underset{{k}} {\overset{\mathrm{2014}} {\:}}\right)\:\centerdot\:\mathrm{3}^{{k}} \:. \\ $$

Question Number 162071    Answers: 1   Comments: 9

Question Number 162068    Answers: 1   Comments: 0

Question Number 162066    Answers: 0   Comments: 0

Ξ£_(n=1) ^∞ (((βˆ’1)^n H_n )/n^2 )=???

$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(βˆ’\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}^{\mathrm{2}} }=??? \\ $$

  Pg 559      Pg 560      Pg 561      Pg 562      Pg 563      Pg 564      Pg 565      Pg 566      Pg 567      Pg 568   

Terms of Service

Privacy Policy

Contact: info@tinkutara.com