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Question Number 85603    Answers: 0   Comments: 0

prove the relation ∫_0 ^1 ((li_5 ((x)^(1/5) ))/(x)^(1/5) )dx=(5/4)(((25)/(3072))−((ζ(2))/2^6 )+((ζ(3))/2^4 )−((ζ(4))/2^2 )+ζ(5))

$${prove}\:{the}\:{relation} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{li}_{\mathrm{5}} \left(\sqrt[{\mathrm{5}}]{{x}}\right)}{\sqrt[{\mathrm{5}}]{{x}}}{dx}=\frac{\mathrm{5}}{\mathrm{4}}\left(\frac{\mathrm{25}}{\mathrm{3072}}−\frac{\zeta\left(\mathrm{2}\right)}{\mathrm{2}^{\mathrm{6}} }+\frac{\zeta\left(\mathrm{3}\right)}{\mathrm{2}^{\mathrm{4}} }−\frac{\zeta\left(\mathrm{4}\right)}{\mathrm{2}^{\mathrm{2}} }+\zeta\left(\mathrm{5}\right)\right) \\ $$

Question Number 85592    Answers: 1   Comments: 0

∫(((u+1)^2 )/(u^3 +u))du

$$\int\frac{\left(\mathrm{u}+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{u}^{\mathrm{3}} +\mathrm{u}}\mathrm{du} \\ $$

Question Number 85591    Answers: 1   Comments: 0

∫((1+4u)/(−4u^2 +2u+2))du

$$\int\frac{\mathrm{1}+\mathrm{4u}}{−\mathrm{4u}^{\mathrm{2}} +\mathrm{2u}+\mathrm{2}}\mathrm{du} \\ $$$$ \\ $$

Question Number 85590    Answers: 0   Comments: 0

∫((1+4u)/(−4u^2 +2u+2))du

$$\int\frac{\mathrm{1}+\mathrm{4u}}{−\mathrm{4u}^{\mathrm{2}} +\mathrm{2u}+\mathrm{2}}\mathrm{du} \\ $$$$ \\ $$

Question Number 85588    Answers: 0   Comments: 6

Question Number 85601    Answers: 0   Comments: 2

∫((4u)/(4u^2 −4u+1))du

$$\int\frac{\mathrm{4u}}{\mathrm{4u}^{\mathrm{2}} −\mathrm{4u}+\mathrm{1}}\mathrm{du} \\ $$

Question Number 85600    Answers: 1   Comments: 3

∫(x^2 /(√(1+x^2 ))) dx

$$\int\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 85596    Answers: 1   Comments: 1

∫(((√(x+1))−1)/((√(x−1))+1)) dx

$$\int\frac{\sqrt{{x}+\mathrm{1}}−\mathrm{1}}{\sqrt{{x}−\mathrm{1}}+\mathrm{1}}\:{dx} \\ $$

Question Number 85583    Answers: 0   Comments: 0

Question Number 85580    Answers: 0   Comments: 0

Solve: (D^2 +2D+1)y= x cos x

$$\boldsymbol{\mathrm{Solve}}: \\ $$$$\:\left(\mathrm{D}^{\mathrm{2}} +\mathrm{2D}+\mathrm{1}\right)\mathrm{y}=\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x} \\ $$$$ \\ $$

Question Number 85582    Answers: 0   Comments: 1

cos ((π/9))+cos (((2π)/9))+cos (((4π)/9))=

$$\mathrm{cos}\:\left(\frac{\pi}{\mathrm{9}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{9}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{9}}\right)= \\ $$

Question Number 85568    Answers: 4   Comments: 2

∫ _0 ^(2π) (dx/((√2)−cos x))

$$\int\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\:}}\:\frac{\mathrm{dx}}{\sqrt{\mathrm{2}}−\mathrm{cos}\:\mathrm{x}} \\ $$

Question Number 85557    Answers: 0   Comments: 3

x = (√(1+ (√(5+ (√(11+ (√(19+...)))))))) x = ?

$${x}\:\:=\:\:\sqrt{\mathrm{1}+\:\sqrt{\mathrm{5}+\:\sqrt{\mathrm{11}+\:\sqrt{\mathrm{19}+...}}}} \\ $$$${x}\:\:=\:\:\:? \\ $$

Question Number 85555    Answers: 1   Comments: 0

2x^2 +5x+7=0

$$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{7}=\mathrm{0} \\ $$

Question Number 85554    Answers: 0   Comments: 1

2x^2 +5x+7=0

$$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{7}=\mathrm{0} \\ $$

Question Number 85551    Answers: 0   Comments: 1

∫ (dx/(x^2 (x^4 +1)^(3/4) ))

$$\int\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{4}} +\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{4}}} } \\ $$

Question Number 85546    Answers: 0   Comments: 1

Question Number 85542    Answers: 1   Comments: 4

Question Number 85540    Answers: 1   Comments: 0

find the range y=((x+[x])/(1−[x]+x))

$${find}\:{the}\:{range} \\ $$$${y}=\frac{{x}+\left[{x}\right]}{\mathrm{1}−\left[{x}\right]+{x}} \\ $$

Question Number 85535    Answers: 0   Comments: 0

Question Number 85534    Answers: 1   Comments: 2

Question Number 85532    Answers: 1   Comments: 0

Find the term independent of x in the expression of (2x−(1/(2x)))^9

$${Find}\:{the}\:{term}\:{independent}\:{of}\:\boldsymbol{\mathrm{x}}\:{in}\:{the}\:{expression}\:{of}\:\left(\mathrm{2}{x}−\frac{\mathrm{1}}{\mathrm{2}{x}}\right)^{\mathrm{9}} \\ $$

Question Number 85523    Answers: 1   Comments: 1

Question Number 85589    Answers: 0   Comments: 0

∫((1+4u)/(−4u^2 +2u+2))du

$$\int\frac{\mathrm{1}+\mathrm{4u}}{−\mathrm{4u}^{\mathrm{2}} +\mathrm{2u}+\mathrm{2}}\mathrm{du} \\ $$$$ \\ $$

Question Number 85503    Answers: 1   Comments: 0

(dy/dx) = sec (x+y)

$$\frac{{dy}}{{dx}}\:=\:\mathrm{sec}\:\left({x}+{y}\right)\: \\ $$

Question Number 85500    Answers: 1   Comments: 0

The function of f and g are defined by f:g→(x/(bx−2)), x ≠ (2/b) and b ≠ 0, where a and b are real numbers g:x →2x−11 (a) If f(2)= ((-1)/2) and f^(−1) (1) = -1, find a and b and write down the expression for f in terms of x (b) Find the value of x for which fg(x)= ((-1)/2)

$${The}\:{function}\:{of}\:\boldsymbol{\mathrm{f}}\:{and}\:\boldsymbol{\mathrm{g}}\:{are}\:{defined}\:{by}\:\boldsymbol{\mathrm{f}}:\boldsymbol{\mathrm{g}}\rightarrow\frac{{x}}{\boldsymbol{\mathrm{b}}{x}−\mathrm{2}},\:{x}\:\neq\:\frac{\mathrm{2}}{\boldsymbol{\mathrm{b}}}\:{and}\:\boldsymbol{\mathrm{b}}\:\neq\:\mathrm{0},\:{where}\:\boldsymbol{\mathrm{a}}\:{and}\:\boldsymbol{\mathrm{b}}\:{are}\:{real}\:{numbers}\:\boldsymbol{\mathrm{g}}:\boldsymbol{\mathrm{x}}\:\rightarrow\mathrm{2}{x}−\mathrm{11} \\ $$$$\left({a}\right)\:{If}\:\boldsymbol{\mathrm{f}}\left(\mathrm{2}\right)=\:\frac{-\mathrm{1}}{\mathrm{2}}\:{and}\:\boldsymbol{\mathrm{f}}^{−\mathrm{1}} \left(\mathrm{1}\right)\:=\:-\mathrm{1},\:{find}\:\boldsymbol{\mathrm{a}}\:{and}\:\boldsymbol{\mathrm{b}}\:{and}\:{write}\:{down}\:{the}\:{expression}\:{for}\:\boldsymbol{\mathrm{f}}\:{in}\:{terms}\:{of}\:\boldsymbol{\mathrm{x}} \\ $$$$\left(\boldsymbol{\mathrm{b}}\right)\:\boldsymbol{\mathrm{F}}{ind}\:{the}\:{value}\:{of}\:\boldsymbol{\mathrm{x}}\:{for}\:{which}\:\boldsymbol{\mathrm{fg}}\left(\boldsymbol{\mathrm{x}}\right)=\:\frac{-\mathrm{1}}{\mathrm{2}} \\ $$

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