Elementary Functions - 4.26 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.26.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\sin@@{x}\diff{x} = -\cos@@{x}}
\int\sin@@{x}\diff{x} = -\cos@@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(sin(x), x) = - cos(x)

Integrate[Sin[x], x, GenerateConditions->None] == - Cos[x]
Successful Successful - Successful [Tested: 3]
4.26.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\cos@@{x}\diff{x} = \sin@@{x}}
\int\cos@@{x}\diff{x} = \sin@@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(cos(x), x) = sin(x)

Integrate[Cos[x], x, GenerateConditions->None] == Sin[x]
Successful Successful - Successful [Tested: 3]
4.26.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\tan@@{x}\diff{x} = -\ln@{\cos@@{x}}}
\int\tan@@{x}\diff{x} = -\ln@{\cos@@{x}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\tfrac{1}{2}\pi < x, x < \tfrac{1}{2}\pi} 
int(tan(x), x) = - ln(cos(x))

Integrate[Tan[x], x, GenerateConditions->None] == - Log[Cos[x]]
Successful Successful - Successful [Tested: 2]
4.26.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\csc@@{x}\diff{x} = \ln@{\tan@@{\tfrac{1}{2}x}}}
\int\csc@@{x}\diff{x} = \ln@{\tan@@{\tfrac{1}{2}x}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < x, x < \pi} 
int(csc(x), x) = ln(tan((1)/(2)*x))

Integrate[Csc[x], x, GenerateConditions->None] == Log[Tan[Divide[1,2]*x]]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
4.26.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\sec@@{x}\diff{x} = \aGudermannian@{x}}
\int\sec@@{x}\diff{x} = \aGudermannian@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\frac{1}{2}\pi < x, x < \frac{1}{2}\pi} 
int(sec(x), x) = arctanh(sin(x))

Integrate[Sec[x], x, GenerateConditions->None] == InverseGudermannian[x]
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
4.26.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\cot@@{x}\diff{x} = \ln@{\sin@@{x}}}
\int\cot@@{x}\diff{x} = \ln@{\sin@@{x}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < x, x < \pi} 
int(cot(x), x) = ln(sin(x))

Integrate[Cot[x], x, GenerateConditions->None] == Log[Sin[x]]
Successful Successful - Successful [Tested: 3]
4.26.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int e^{ax}\sin@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\sin@{bx}-b\cos@{bx})}
\int e^{ax}\sin@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\sin@{bx}-b\cos@{bx})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(exp(a*x)*sin(b*x), x) = (exp(a*x))/((a)^(2)+ (b)^(2))*(a*sin(b*x)- b*cos(b*x))

Integrate[Exp[a*x]*Sin[b*x], x, GenerateConditions->None] == Divide[Exp[a*x],(a)^(2)+ (b)^(2)]*(a*Sin[b*x]- b*Cos[b*x])
Successful Successful - Successful [Tested: 108]
4.26.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int e^{ax}\cos@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\cos@{bx}+b\sin@{bx})}
\int e^{ax}\cos@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\cos@{bx}+b\sin@{bx})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(exp(a*x)*cos(b*x), x) = (exp(a*x))/((a)^(2)+ (b)^(2))*(a*cos(b*x)+ b*sin(b*x))

Integrate[Exp[a*x]*Cos[b*x], x, GenerateConditions->None] == Divide[Exp[a*x],(a)^(2)+ (b)^(2)]*(a*Cos[b*x]+ b*Sin[b*x])
Successful Successful - Successful [Tested: 108]
4.26.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\pi}\sin@{mt}\sin@{nt}\diff{t} = 0}
\int_{0}^{\pi}\sin@{mt}\sin@{nt}\diff{t} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m \neq n} 
int(sin(m*t)*sin(n*t), t = 0..Pi) = 0

Integrate[Sin[m*t]*Sin[n*t], {t, 0, Pi}, GenerateConditions->None] == 0
Successful Failure - Successful [Tested: 6]
4.26.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\pi}\cos@{mt}\cos@{nt}\diff{t} = 0}
\int_{0}^{\pi}\cos@{mt}\cos@{nt}\diff{t} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m \neq n} 
int(cos(m*t)*cos(n*t), t = 0..Pi) = 0

Integrate[Cos[m*t]*Cos[n*t], {t, 0, Pi}, GenerateConditions->None] == 0
Successful Failure - Successful [Tested: 6]
4.26.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\pi}\sin^{2}@{nt}\diff{t} = \int_{0}^{\pi}\cos^{2}@{nt}\diff{t}}
\int_{0}^{\pi}\sin^{2}@{nt}\diff{t} = \int_{0}^{\pi}\cos^{2}@{nt}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \neq 0} 
int((sin(n*t))^(2), t = 0..Pi) = int((cos(n*t))^(2), t = 0..Pi)

Integrate[(Sin[n*t])^(2), {t, 0, Pi}, GenerateConditions->None] == Integrate[(Cos[n*t])^(2), {t, 0, Pi}, GenerateConditions->None]
Successful Failure - Successful [Tested: 3]
4.26.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\pi}\cos^{2}@{nt}\diff{t} = \tfrac{1}{2}\pi}
\int_{0}^{\pi}\cos^{2}@{nt}\diff{t} = \tfrac{1}{2}\pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \neq 0} 
int((cos(n*t))^(2), t = 0..Pi) = (1)/(2)*Pi

Integrate[(Cos[n*t])^(2), {t, 0, Pi}, GenerateConditions->None] == Divide[1,2]*Pi
Successful Failure - Successful [Tested: 3]
4.26.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\sin@{t^{2}}\diff{t} = \int_{0}^{\infty}\cos@{t^{2}}\diff{t}}
\int_{0}^{\infty}\sin@{t^{2}}\diff{t} = \int_{0}^{\infty}\cos@{t^{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(sin((t)^(2)), t = 0..infinity) = int(cos((t)^(2)), t = 0..infinity)

Integrate[Sin[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 1]
4.26.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\cos@{t^{2}}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{2}}}
\int_{0}^{\infty}\cos@{t^{2}}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(cos((t)^(2)), t = 0..infinity) = (1)/(2)*sqrt((Pi)/(2))

Integrate[Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Sqrt[Divide[Pi,2]]
Successful Successful - Successful [Tested: 1]
4.26.E14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\asin@@{x}\diff{x} = x\asin@@{x}+(1-x^{2})^{1/2}}
\int\asin@@{x}\diff{x} = x\asin@@{x}+(1-x^{2})^{1/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -1 < x, x < 1} 
int(arcsin(x), x) = x*arcsin(x)+(1 - (x)^(2))^(1/2)

Integrate[ArcSin[x], x, GenerateConditions->None] == x*ArcSin[x]+(1 - (x)^(2))^(1/2)
Successful Successful - Successful [Tested: 1]
4.26.E15 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\acos@@{x}\diff{x} = x\acos@@{x}-(1-x^{2})^{1/2}}
\int\acos@@{x}\diff{x} = x\acos@@{x}-(1-x^{2})^{1/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -1 < x, x < 1} 
int(arccos(x), x) = x*arccos(x)-(1 - (x)^(2))^(1/2)

Integrate[ArcCos[x], x, GenerateConditions->None] == x*ArcCos[x]-(1 - (x)^(2))^(1/2)
Successful Successful - Successful [Tested: 1]
4.26.E16 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\atan@@{x}\diff{x} = x\atan@@{x}-\tfrac{1}{2}\ln@{1+x^{2}}}
\int\atan@@{x}\diff{x} = x\atan@@{x}-\tfrac{1}{2}\ln@{1+x^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\infty < x, x < \infty} 
int(arctan(x), x) = x*arctan(x)-(1)/(2)*ln(1 + (x)^(2))

Integrate[ArcTan[x], x, GenerateConditions->None] == x*ArcTan[x]-Divide[1,2]*Log[1 + (x)^(2)]
Successful Successful - Successful [Tested: 3]
4.26.E17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\acsc@@{x}\diff{x} = x\acsc@@{x}+\ln@{x+(x^{2}-1)^{1/2}}}
\int\acsc@@{x}\diff{x} = x\acsc@@{x}+\ln@{x+(x^{2}-1)^{1/2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1 < x, x < \infty} 
int(arccsc(x), x) = x*arccsc(x)+ ln(x +((x)^(2)- 1)^(1/2))

Integrate[ArcCsc[x], x, GenerateConditions->None] == x*ArcCsc[x]+ Log[x +((x)^(2)- 1)^(1/2)]
Successful Failure -
Failed [2 / 2]
Result: Complex[-1.1102230246251565*^-16, -1.5707963267948966]
Test Values: {Rule[x, 1.5]}


Result: Complex[-4.440892098500626*^-16, -1.5707963267948966]
Test Values: {Rule[x, 2]}

4.26.E18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\asec@@{x}\diff{x} = x\asec@@{x}-\ln@{x+(x^{2}-1)^{1/2}}}
\int\asec@@{x}\diff{x} = x\asec@@{x}-\ln@{x+(x^{2}-1)^{1/2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1 < x, x < \infty} 
int(arcsec(x), x) = x*arcsec(x)- ln(x +((x)^(2)- 1)^(1/2))

Integrate[ArcSec[x], x, GenerateConditions->None] == x*ArcSec[x]- Log[x +((x)^(2)- 1)^(1/2)]
Successful Failure -
Failed [2 / 2]
Result: Complex[1.1102230246251565*^-16, 1.5707963267948966]
Test Values: {Rule[x, 1.5]}


Result: Complex[4.440892098500626*^-16, 1.5707963267948966]
Test Values: {Rule[x, 2]}

4.26.E19 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\acot@@{x}\diff{x} = x\acot@@{x}+\tfrac{1}{2}\ln@{1+x^{2}}}
\int\acot@@{x}\diff{x} = x\acot@@{x}+\tfrac{1}{2}\ln@{1+x^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < x, x < \infty} 
int(arccot(x), x) = x*arccot(x)+(1)/(2)*ln(1 + (x)^(2))

Integrate[ArcCot[x], x, GenerateConditions->None] == x*ArcCot[x]+Divide[1,2]*Log[1 + (x)^(2)]
Successful Successful - Successful [Tested: 3]
4.26.E20 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int x\asin@@{x}\diff{x} = \left(\frac{x^{2}}{2}-\frac{1}{4}\right)\asin@@{x}+\frac{x}{4}(1-x^{2})^{1/2}}
\int x\asin@@{x}\diff{x} = \left(\frac{x^{2}}{2}-\frac{1}{4}\right)\asin@@{x}+\frac{x}{4}(1-x^{2})^{1/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -1 < x, x < 1} 
int(x*arcsin(x), x) = (((x)^(2))/(2)-(1)/(4))*arcsin(x)+(x)/(4)*(1 - (x)^(2))^(1/2)

Integrate[x*ArcSin[x], x, GenerateConditions->None] == (Divide[(x)^(2),2]-Divide[1,4])*ArcSin[x]+Divide[x,4]*(1 - (x)^(2))^(1/2)
Successful Successful - Successful [Tested: 1]
4.26.E21 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int x\acos@@{x}\diff{x} = \left(\frac{x^{2}}{2}-\frac{1}{4}\right)\acos@@{x}-\frac{x}{4}(1-x^{2})^{1/2}}
\int x\acos@@{x}\diff{x} = \left(\frac{x^{2}}{2}-\frac{1}{4}\right)\acos@@{x}-\frac{x}{4}(1-x^{2})^{1/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -1 < x, x < 1} 
int(x*arccos(x), x) = (((x)^(2))/(2)-(1)/(4))*arccos(x)-(x)/(4)*(1 - (x)^(2))^(1/2)

Integrate[x*ArcCos[x], x, GenerateConditions->None] == (Divide[(x)^(2),2]-Divide[1,4])*ArcCos[x]-Divide[x,4]*(1 - (x)^(2))^(1/2)
Failure Failure
Failed [1 / 1]
Result: .3926990817
Test Values: {x = .5}

Failed [1 / 1]
Result: 0.3926990816987242
Test Values: {Rule[x, 0.5]}

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