View source for 4.26

{{DISPLAYTITLE:Elementary Functions - 4.26 Integrals}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/4.26.E1 4.26.E1] || [[Item:Q1817|<math>\int\sin@@{x}\diff{x} = -\cos@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\sin@@{x}\diff{x} = -\cos@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(sin(x), x) = - cos(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Sin[x], x, GenerateConditions->None] == - Cos[x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.26.E2 4.26.E2] || [[Item:Q1818|<math>\int\cos@@{x}\diff{x} = \sin@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\cos@@{x}\diff{x} = \sin@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(cos(x), x) = sin(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Cos[x], x, GenerateConditions->None] == Sin[x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.26.E3 4.26.E3] || [[Item:Q1819|<math>\int\tan@@{x}\diff{x} = -\ln@{\cos@@{x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\tan@@{x}\diff{x} = -\ln@{\cos@@{x}}</syntaxhighlight> || <math>-\tfrac{1}{2}\pi < x, x < \tfrac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>int(tan(x), x) = - ln(cos(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Tan[x], x, GenerateConditions->None] == - Log[Cos[x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.26.E4 4.26.E4] || [[Item:Q1820|<math>\int\csc@@{x}\diff{x} = \ln@{\tan@@{\tfrac{1}{2}x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\csc@@{x}\diff{x} = \ln@{\tan@@{\tfrac{1}{2}x}}</syntaxhighlight> || <math>0 < x, x < \pi</math> || <syntaxhighlight lang=mathematica>int(csc(x), x) = ln(tan((1)/(2)*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Csc[x], x, GenerateConditions->None] == Log[Tan[Divide[1,2]*x]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.26.E5 4.26.E5] || [[Item:Q1821|<math>\int\sec@@{x}\diff{x} = \aGudermannian@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\sec@@{x}\diff{x} = \aGudermannian@{x}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>int(sec(x), x) = arctanh(sin(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Sec[x], x, GenerateConditions->None] == InverseGudermannian[x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.26.E6 4.26.E6] || [[Item:Q1822|<math>\int\cot@@{x}\diff{x} = \ln@{\sin@@{x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\cot@@{x}\diff{x} = \ln@{\sin@@{x}}</syntaxhighlight> || <math>0 < x, x < \pi</math> || <syntaxhighlight lang=mathematica>int(cot(x), x) = ln(sin(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Cot[x], x, GenerateConditions->None] == Log[Sin[x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.26.E7 4.26.E7] || [[Item:Q1823|<math>\int e^{ax}\sin@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\sin@{bx}-b\cos@{bx})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int e^{ax}\sin@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\sin@{bx}-b\cos@{bx})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(exp(a*x)*sin(b*x), x) = (exp(a*x))/((a)^(2)+ (b)^(2))*(a*sin(b*x)- b*cos(b*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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])</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 108]
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| [https://dlmf.nist.gov/4.26.E8 4.26.E8] || [[Item:Q1824|<math>\int e^{ax}\cos@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\cos@{bx}+b\sin@{bx})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int e^{ax}\cos@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\cos@{bx}+b\sin@{bx})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(exp(a*x)*cos(b*x), x) = (exp(a*x))/((a)^(2)+ (b)^(2))*(a*cos(b*x)+ b*sin(b*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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])</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 108]
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| [https://dlmf.nist.gov/4.26.E9 4.26.E9] || [[Item:Q1825|<math>\int_{0}^{\pi}\sin@{mt}\sin@{nt}\diff{t} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi}\sin@{mt}\sin@{nt}\diff{t} = 0</syntaxhighlight> || <math>m \neq n</math> || <syntaxhighlight lang=mathematica>int(sin(m*t)*sin(n*t), t = 0..Pi) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Sin[m*t]*Sin[n*t], {t, 0, Pi}, GenerateConditions->None] == 0</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 6]
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| [https://dlmf.nist.gov/4.26.E10 4.26.E10] || [[Item:Q1826|<math>\int_{0}^{\pi}\cos@{mt}\cos@{nt}\diff{t} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi}\cos@{mt}\cos@{nt}\diff{t} = 0</syntaxhighlight> || <math>m \neq n</math> || <syntaxhighlight lang=mathematica>int(cos(m*t)*cos(n*t), t = 0..Pi) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Cos[m*t]*Cos[n*t], {t, 0, Pi}, GenerateConditions->None] == 0</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 6]
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| [https://dlmf.nist.gov/4.26.E11 4.26.E11] || [[Item:Q1827|<math>\int_{0}^{\pi}\sin^{2}@{nt}\diff{t} = \int_{0}^{\pi}\cos^{2}@{nt}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi}\sin^{2}@{nt}\diff{t} = \int_{0}^{\pi}\cos^{2}@{nt}\diff{t}</syntaxhighlight> || <math>n \neq 0</math> || <syntaxhighlight lang=mathematica>int((sin(n*t))^(2), t = 0..Pi) = int((cos(n*t))^(2), t = 0..Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(Sin[n*t])^(2), {t, 0, Pi}, GenerateConditions->None] == Integrate[(Cos[n*t])^(2), {t, 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.26.E11 4.26.E11] || [[Item:Q1827|<math>\int_{0}^{\pi}\cos^{2}@{nt}\diff{t} = \tfrac{1}{2}\pi</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi}\cos^{2}@{nt}\diff{t} = \tfrac{1}{2}\pi</syntaxhighlight> || <math>n \neq 0</math> || <syntaxhighlight lang=mathematica>int((cos(n*t))^(2), t = 0..Pi) = (1)/(2)*Pi</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(Cos[n*t])^(2), {t, 0, Pi}, GenerateConditions->None] == Divide[1,2]*Pi</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.26.E13 4.26.E13] || [[Item:Q1829|<math>\int_{0}^{\infty}\sin@{t^{2}}\diff{t} = \int_{0}^{\infty}\cos@{t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\sin@{t^{2}}\diff{t} = \int_{0}^{\infty}\cos@{t^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(sin((t)^(2)), t = 0..infinity) = int(cos((t)^(2)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Sin[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.26.E13 4.26.E13] || [[Item:Q1829|<math>\int_{0}^{\infty}\cos@{t^{2}}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\cos@{t^{2}}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(cos((t)^(2)), t = 0..infinity) = (1)/(2)*sqrt((Pi)/(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Sqrt[Divide[Pi,2]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.26.E14 4.26.E14] || [[Item:Q1830|<math>\int\asin@@{x}\diff{x} = x\asin@@{x}+(1-x^{2})^{1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\asin@@{x}\diff{x} = x\asin@@{x}+(1-x^{2})^{1/2}</syntaxhighlight> || <math>-1 < x, x < 1</math> || <syntaxhighlight lang=mathematica>int(arcsin(x), x) = x*arcsin(x)+(1 - (x)^(2))^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcSin[x], x, GenerateConditions->None] == x*ArcSin[x]+(1 - (x)^(2))^(1/2)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.26.E15 4.26.E15] || [[Item:Q1831|<math>\int\acos@@{x}\diff{x} = x\acos@@{x}-(1-x^{2})^{1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\acos@@{x}\diff{x} = x\acos@@{x}-(1-x^{2})^{1/2}</syntaxhighlight> || <math>-1 < x, x < 1</math> || <syntaxhighlight lang=mathematica>int(arccos(x), x) = x*arccos(x)-(1 - (x)^(2))^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcCos[x], x, GenerateConditions->None] == x*ArcCos[x]-(1 - (x)^(2))^(1/2)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.26.E16 4.26.E16] || [[Item:Q1832|<math>\int\atan@@{x}\diff{x} = x\atan@@{x}-\tfrac{1}{2}\ln@{1+x^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\atan@@{x}\diff{x} = x\atan@@{x}-\tfrac{1}{2}\ln@{1+x^{2}}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(arctan(x), x) = x*arctan(x)-(1)/(2)*ln(1 + (x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcTan[x], x, GenerateConditions->None] == x*ArcTan[x]-Divide[1,2]*Log[1 + (x)^(2)]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.26.E17 4.26.E17] || [[Item:Q1833|<math>\int\acsc@@{x}\diff{x} = x\acsc@@{x}+\ln@{x+(x^{2}-1)^{1/2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\acsc@@{x}\diff{x} = x\acsc@@{x}+\ln@{x+(x^{2}-1)^{1/2}}</syntaxhighlight> || <math>1 < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(arccsc(x), x) = x*arccsc(x)+ ln(x +((x)^(2)- 1)^(1/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcCsc[x], x, GenerateConditions->None] == x*ArcCsc[x]+ Log[x +((x)^(2)- 1)^(1/2)]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 2]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.1102230246251565*^-16, -1.5707963267948966]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-4.440892098500626*^-16, -1.5707963267948966]
Test Values: {Rule[x, 2]}</syntaxhighlight><br></div></div>
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| [https://dlmf.nist.gov/4.26.E18 4.26.E18] || [[Item:Q1834|<math>\int\asec@@{x}\diff{x} = x\asec@@{x}-\ln@{x+(x^{2}-1)^{1/2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\asec@@{x}\diff{x} = x\asec@@{x}-\ln@{x+(x^{2}-1)^{1/2}}</syntaxhighlight> || <math>1 < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(arcsec(x), x) = x*arcsec(x)- ln(x +((x)^(2)- 1)^(1/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcSec[x], x, GenerateConditions->None] == x*ArcSec[x]- Log[x +((x)^(2)- 1)^(1/2)]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 2]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.1102230246251565*^-16, 1.5707963267948966]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[4.440892098500626*^-16, 1.5707963267948966]
Test Values: {Rule[x, 2]}</syntaxhighlight><br></div></div>
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| [https://dlmf.nist.gov/4.26.E19 4.26.E19] || [[Item:Q1835|<math>\int\acot@@{x}\diff{x} = x\acot@@{x}+\tfrac{1}{2}\ln@{1+x^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\acot@@{x}\diff{x} = x\acot@@{x}+\tfrac{1}{2}\ln@{1+x^{2}}</syntaxhighlight> || <math>0 < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(arccot(x), x) = x*arccot(x)+(1)/(2)*ln(1 + (x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcCot[x], x, GenerateConditions->None] == x*ArcCot[x]+Divide[1,2]*Log[1 + (x)^(2)]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.26.E20 4.26.E20] || [[Item:Q1836|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>-1 < x, x < 1</math> || <syntaxhighlight lang=mathematica>int(x*arcsin(x), x) = (((x)^(2))/(2)-(1)/(4))*arcsin(x)+(x)/(4)*(1 - (x)^(2))^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[x*ArcSin[x], x, GenerateConditions->None] == (Divide[(x)^(2),2]-Divide[1,4])*ArcSin[x]+Divide[x,4]*(1 - (x)^(2))^(1/2)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.26.E21 4.26.E21] || [[Item:Q1837|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>-1 < x, x < 1</math> || <syntaxhighlight lang=mathematica>int(x*arccos(x), x) = (((x)^(2))/(2)-(1)/(4))*arccos(x)-(x)/(4)*(1 - (x)^(2))^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[x*ArcCos[x], x, GenerateConditions->None] == (Divide[(x)^(2),2]-Divide[1,4])*ArcCos[x]-Divide[x,4]*(1 - (x)^(2))^(1/2)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3926990817
Test Values: {x = .5}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0.3926990816987242
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br></div></div>
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