Elementary Functions - 4.10 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.10.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\frac{\diff{z}}{z} = \ln@@{z}}
\int\frac{\diff{z}}{z} = \ln@@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((1)/(z), z) = ln(z)

Integrate[Divide[1,z], z, GenerateConditions->None] == Log[z]
Successful Successful - Successful [Tested: 7]
4.10.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\ln@@{z}\diff{z} = z\ln@@{z}-z}
\int\ln@@{z}\diff{z} = z\ln@@{z}-z		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(ln(z), z) = z*ln(z)- z

Integrate[Log[z], z, GenerateConditions->None] == z*Log[z]- z
Successful Successful - Successful [Tested: 7]
4.10.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int z^{n}\ln@@{z}\diff{z} = \frac{z^{n+1}}{n+1}\ln@@{z}-\frac{z^{n+1}}{(n+1)^{2}}}
\int z^{n}\ln@@{z}\diff{z} = \frac{z^{n+1}}{n+1}\ln@@{z}-\frac{z^{n+1}}{(n+1)^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \neq -1} 
int((z)^(n)* ln(z), z) = ((z)^(n + 1))/(n + 1)*ln(z)-((z)^(n + 1))/((n + 1)^(2))

Integrate[(z)^(n)* Log[z], z, GenerateConditions->None] == Divide[(z)^(n + 1),n + 1]*Log[z]-Divide[(z)^(n + 1),(n + 1)^(2)]
Successful Successful - Successful [Tested: 21]
4.10.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\frac{\diff{z}}{z\ln@@{z}} = \ln@{\ln@@{z}}}
\int\frac{\diff{z}}{z\ln@@{z}} = \ln@{\ln@@{z}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((1)/(z*ln(z)), z) = ln(ln(z))

Integrate[Divide[1,z*Log[z]], z, GenerateConditions->None] == Log[Log[z]]
Successful Successful - Successful [Tested: 7]
4.10.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\frac{\ln@@{t}}{1-t}\diff{t} = -\frac{\pi^{2}}{6}}
\int_{0}^{1}\frac{\ln@@{t}}{1-t}\diff{t} = -\frac{\pi^{2}}{6}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((ln(t))/(1 - t), t = 0..1) = -((Pi)^(2))/(6)

Integrate[Divide[Log[t],1 - t], {t, 0, 1}, GenerateConditions->None] == -Divide[(Pi)^(2),6]
Successful Successful - Successful [Tested: 1]
4.10.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\frac{\ln@@{t}}{1+t}\diff{t} = -\frac{\pi^{2}}{12}}
\int_{0}^{1}\frac{\ln@@{t}}{1+t}\diff{t} = -\frac{\pi^{2}}{12}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((ln(t))/(1 + t), t = 0..1) = -((Pi)^(2))/(12)

Integrate[Divide[Log[t],1 + t], {t, 0, 1}, GenerateConditions->None] == -Divide[(Pi)^(2),12]
Successful Successful - Successful [Tested: 1]
4.10.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int e^{az}\diff{z} = \frac{e^{az}}{a}}
\int e^{az}\diff{z} = \frac{e^{az}}{a}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(exp(a*z), z) = (exp(a*z))/(a)

Integrate[Exp[a*z], z, GenerateConditions->None] == Divide[Exp[a*z],a]
Successful Successful - Successful [Tested: 42]
4.10.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\frac{\diff{z}}{e^{az}+b} = \frac{1}{ab}(az-\ln@{e^{az}+b})}
\int\frac{\diff{z}}{e^{az}+b} = \frac{1}{ab}(az-\ln@{e^{az}+b})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((1)/(exp(a*z)+ b), z) = (1)/(a*b)*(a*z - ln(exp(a*z)+ b))

Integrate[Divide[1,Exp[a*z]+ b], z, GenerateConditions->None] == Divide[1,a*b]*(a*z - Log[Exp[a*z]+ b])
Failure Successful Successful [Tested: 252] Successful [Tested: 252]
4.10.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\frac{e^{az}-1}{e^{az}+1}\diff{z} = \frac{2}{a}\ln@{e^{az/2}+e^{-az/2}}}
\int\frac{e^{az}-1}{e^{az}+1}\diff{z} = \frac{2}{a}\ln@{e^{az/2}+e^{-az/2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((exp(a*z)- 1)/(exp(a*z)+ 1), z) = (2)/(a)*ln(exp(a*z/2)+ exp(- a*z/2))

Integrate[Divide[Exp[a*z]- 1,Exp[a*z]+ 1], z, GenerateConditions->None] == Divide[2,a]*Log[Exp[a*z/2]+ Exp[- a*z/2]]
Failure Failure Successful [Tested: 42] Successful [Tested: 42]
4.10.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-\infty}^{\infty}e^{-cx^{2}}\diff{x} = \sqrt{\frac{\pi}{c}}}
\int_{-\infty}^{\infty}e^{-cx^{2}}\diff{x} = \sqrt{\frac{\pi}{c}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{c} > 0} 
int(exp(- c*(x)^(2)), x = - infinity..infinity) = sqrt((Pi)/(c))

Integrate[Exp[- c*(x)^(2)], {x, - Infinity, Infinity}, GenerateConditions->None] == Sqrt[Divide[Pi,c]]
Successful Successful - Successful [Tested: 3]
4.10.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\ln@@{2}}\frac{xe^{x}}{e^{x}-1}\diff{x} = \frac{\pi^{2}}{12}}
\int_{0}^{\ln@@{2}}\frac{xe^{x}}{e^{x}-1}\diff{x} = \frac{\pi^{2}}{12}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((x*exp(x))/(exp(x)- 1), x = 0..ln(2)) = ((Pi)^(2))/(12)

Integrate[Divide[x*Exp[x],Exp[x]- 1], {x, 0, Log[2]}, GenerateConditions->None] == Divide[(Pi)^(2),12]
Successful Successful - Successful [Tested: 1]
4.10.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\frac{\diff{x}}{e^{x}+1} = \ln@@{2}}
\int_{0}^{\infty}\frac{\diff{x}}{e^{x}+1} = \ln@@{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((1)/(exp(x)+ 1), x = 0..infinity) = ln(2)

Integrate[Divide[1,Exp[x]+ 1], {x, 0, Infinity}, GenerateConditions->None] == Log[2]
Successful Successful - Successful [Tested: 1]
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