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Question Number 103124 Answers: 1 Comments: 0

Li(z)=∫_2 ^z (1/(ln t))dt Li(a+ib)=R(∫_2 ^(a+ib) (1/(ln t))dt)+iI(∫_2 ^(a+ib) (1/(ln t))dt) Can you explain me how get the formula for R(Li(z)) and I(Li(z))?

$$\mathrm{Li}\left({z}\right)=\int_{\mathrm{2}} ^{{z}} \frac{\mathrm{1}}{\mathrm{ln}\:{t}}\mathrm{d}{t} \\ $$$$\mathrm{Li}\left({a}+{ib}\right)=\mathfrak{R}\left(\int_{\mathrm{2}} ^{{a}+{ib}} \frac{\mathrm{1}}{\mathrm{ln}\:{t}}\mathrm{d}{t}\right)+{i}\mathfrak{I}\left(\int_{\mathrm{2}} ^{{a}+{ib}} \frac{\mathrm{1}}{\mathrm{ln}\:{t}}\mathrm{d}{t}\right) \\ $$$$\mathrm{Can}\:\mathrm{you}\:\mathrm{explain}\:\mathrm{me} \\ $$$$\mathrm{how}\:\mathrm{get}\:\mathrm{the}\:\mathrm{formula} \\ $$$$\mathrm{for}\:\mathfrak{R}\left(\mathrm{Li}\left({z}\right)\right)\:\mathrm{and} \\ $$$$\mathfrak{I}\left(\mathrm{Li}\left({z}\right)\right)? \\ $$

Question Number 103120 Answers: 0 Comments: 0

Question Number 103119 Answers: 0 Comments: 0

∫((log(((1+(√5))/2)(√x)−1))/(x^(√x) log(((1+(√5))/2)(√x)+1)−1))

$$\int\frac{{log}\left(\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\sqrt{{x}}−\mathrm{1}\right)}{{x}^{\sqrt{{x}}} {log}\left(\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\sqrt{{x}}+\mathrm{1}\right)−\mathrm{1}} \\ $$

Question Number 103114 Answers: 1 Comments: 2

Question Number 103143 Answers: 1 Comments: 1

Question Number 103094 Answers: 0 Comments: 4

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Question Number 103089 Answers: 1 Comments: 0

solve: (sin^2 (x)−y)dx−tan(x)dy=0

$${solve}: \\ $$$$\left({sin}^{\mathrm{2}} \left({x}\right)−{y}\right){dx}−{tan}\left({x}\right){dy}=\mathrm{0} \\ $$

Question Number 103080 Answers: 0 Comments: 0

calculate A_n =∫_0 ^∞ ((cos(nx))/((1+x^2 )^n ))dx with n integr natural

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({nx}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }{dx} \\ $$$${with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 103078 Answers: 0 Comments: 0

calculate ∫_(−∞) ^∞ ((xsin(2x))/((x^2 +x+1)^2 ))dx

$${calculate}\:\int_{−\infty} ^{\infty} \:\frac{{xsin}\left(\mathrm{2}{x}\right)}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 103077 Answers: 0 Comments: 0

find ∫_0 ^∞ ((x^2 cosx)/((x^2 +x+2)^2 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2}} {cosx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 103067 Answers: 0 Comments: 1

Question Number 103062 Answers: 1 Comments: 1

Question Number 103055 Answers: 0 Comments: 1

Question Number 103047 Answers: 2 Comments: 1

find the product of roots ((2021))^(1/(3 )) x^(log_(2021) (x)) = x^3

$${find}\:{the}\:{product}\:{of}\:{roots}\: \\ $$$$\sqrt[{\mathrm{3}\:}]{\mathrm{2021}}\:{x}^{\mathrm{log}_{\mathrm{2021}} \:\left({x}\right)} \:=\:{x}^{\mathrm{3}} \\ $$

Question Number 103043 Answers: 2 Comments: 0

{ ((x^2 +2xy+21=6x)),((y^2 +2yz−8=6y)),((z^2 +2zx−4=6z)) :} find x+y+z

$$\begin{cases}{{x}^{\mathrm{2}} +\mathrm{2}{xy}+\mathrm{21}=\mathrm{6}{x}}\\{{y}^{\mathrm{2}} +\mathrm{2}{yz}−\mathrm{8}=\mathrm{6}{y}}\\{{z}^{\mathrm{2}} +\mathrm{2}{zx}−\mathrm{4}=\mathrm{6}{z}}\end{cases} \\ $$$${find}\:{x}+{y}+{z}\: \\ $$

Question Number 103040 Answers: 5 Comments: 0

given 5x+12y = 60 min value of (√(x^2 +y^2 ))

$${given}\:\mathrm{5}{x}+\mathrm{12}{y}\:=\:\mathrm{60} \\ $$$${min}\:{value}\:{of}\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} } \\ $$

Question Number 103037 Answers: 1 Comments: 0

4x^2 y′′ +y = 0 , x > 0

$$\mathrm{4x}^{\mathrm{2}} \mathrm{y}''\:+\mathrm{y}\:=\:\mathrm{0}\:,\:\mathrm{x}\:>\:\mathrm{0} \\ $$

Question Number 103021 Answers: 2 Comments: 0

Question Number 103016 Answers: 0 Comments: 0

Question Number 103073 Answers: 1 Comments: 0

I want to write ∫_0 ^(a+ib) f(x)dx but with the form α+iβ How write that? Sorry for my bad english

$$\mathrm{I}\:\mathrm{want}\:\mathrm{to}\:\mathrm{write} \\ $$$$\int_{\mathrm{0}} ^{{a}+{ib}} {f}\left({x}\right)\mathrm{d}{x} \\ $$$$\mathrm{but}\:\mathrm{with}\:\mathrm{the}\:\mathrm{form} \\ $$$$\alpha+{i}\beta \\ $$$$\mathrm{How}\:\mathrm{write}\:\mathrm{that}? \\ $$$$\mathrm{Sorry}\:\mathrm{for}\:\mathrm{my}\:\mathrm{bad} \\ $$$$\mathrm{english} \\ $$

Question Number 103072 Answers: 1 Comments: 9

Question Number 103014 Answers: 2 Comments: 0

y′′ −y = (e^x /(e^x + e^(−x) )) .

$${y}''\:−{y}\:=\:\frac{{e}^{{x}} }{{e}^{{x}} \:+\:{e}^{−{x}} }\:. \\ $$

Question Number 103009 Answers: 2 Comments: 0

(D^2 −4D+4)y = x^3 e^(2x)

$$\left({D}^{\mathrm{2}} −\mathrm{4}{D}+\mathrm{4}\right){y}\:=\:{x}^{\mathrm{3}} {e}^{\mathrm{2}{x}} \: \\ $$

Question Number 103008 Answers: 2 Comments: 0

(dy/dx) − y.tan x = e^x .sec x

$$\frac{{dy}}{{dx}}\:−\:{y}.\mathrm{tan}\:{x}\:=\:{e}^{{x}} .\mathrm{sec}\:{x} \\ $$

Question Number 103004 Answers: 0 Comments: 0

Question Number 103001 Answers: 2 Comments: 0

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