View source for 24.13

{{DISPLAYTITLE:Bernoulli and Euler Polynomials - 24.13 Integrals}}

<div style="width: 100%; height: 75vh; overflow: auto;">
{| class="wikitable sortable" style="margin: 0;"
|-
! 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
|- 
| [https://dlmf.nist.gov/24.13.E2 24.13.E2] || <math qid="Q7536">\int_{x}^{x+1}\BernoullipolyB{n}@{t}\diff{t} = x^{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{x}^{x+1}\BernoullipolyB{n}@{t}\diff{t} = x^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(bernoulli(n, t), t = x..x + 1) = (x)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BernoulliB[n, t], {t, x, x + 1}, GenerateConditions->None] == (x)^(n)</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 9]
|- 
| [https://dlmf.nist.gov/24.13.E3 24.13.E3] || <math qid="Q7537">\int_{x}^{x+(1/2)}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulerpolyE{n}@{2x}}{2^{n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{x}^{x+(1/2)}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulerpolyE{n}@{2x}}{2^{n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(bernoulli(n, t), t = x..x +(1/2)) = (euler(n, 2*x))/((2)^(n + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BernoulliB[n, t], {t, x, x +(1/2)}, GenerateConditions->None] == Divide[EulerE[n, 2*x],(2)^(n + 1)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|- 
| [https://dlmf.nist.gov/24.13.E4 24.13.E4] || <math qid="Q7538">\int_{0}^{1/2}\BernoullipolyB{n}@{t}\diff{t} = \frac{1-2^{n+1}}{2^{n}}\frac{\BernoullinumberB{n+1}}{n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1/2}\BernoullipolyB{n}@{t}\diff{t} = \frac{1-2^{n+1}}{2^{n}}\frac{\BernoullinumberB{n+1}}{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(bernoulli(n, t), t = 0..1/2) = (1 - (2)^(n + 1))/((2)^(n))*(bernoulli(n + 1))/(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BernoulliB[n, t], {t, 0, 1/2}, GenerateConditions->None] == Divide[1 - (2)^(n + 1),(2)^(n)]*Divide[BernoulliB[n + 1],n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/24.13.E5 24.13.E5] || <math qid="Q7539">\int_{1/4}^{3/4}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulernumberE{n}}{2^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{1/4}^{3/4}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulernumberE{n}}{2^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(bernoulli(n, t), t = 1/4..3/4) = (euler(n))/((2)^(2*n + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BernoulliB[n, t], {t, 1/4, 3/4}, GenerateConditions->None] == Divide[EulerE[n],(2)^(2*n + 1)]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/24.13.E6 24.13.E6] || <math qid="Q7540">\int_{0}^{1}\BernoullipolyB{n}@{t}\BernoullipolyB{m}@{t}\diff{t} = \frac{(-1)^{n-1}m!n!}{(m+n)!}\BernoullinumberB{m+n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}\BernoullipolyB{n}@{t}\BernoullipolyB{m}@{t}\diff{t} = \frac{(-1)^{n-1}m!n!}{(m+n)!}\BernoullinumberB{m+n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(bernoulli(n, t)*bernoulli(m, t), t = 0..1) = ((- 1)^(n - 1)* factorial(m)*factorial(n))/(factorial(m + n))*bernoulli(m + n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[BernoulliB[n, t]*BernoulliB[m, t], {t, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n - 1)* (m)!*(n)!,(m + n)!]*BernoulliB[m + n]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|- 
| [https://dlmf.nist.gov/24.13.E8 24.13.E8] || <math qid="Q7542">\int_{0}^{1}\EulerpolyE{n}@{t}\diff{t} = -2\frac{\EulerpolyE{n+1}@{0}}{n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}\EulerpolyE{n}@{t}\diff{t} = -2\frac{\EulerpolyE{n+1}@{0}}{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(euler(n, t), t = 0..1) = - 2*(euler(n + 1, 0))/(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[EulerE[n, t], {t, 0, 1}, GenerateConditions->None] == - 2*Divide[EulerE[n + 1, 0],n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/24.13.E8 24.13.E8] || <math qid="Q7542">-2\frac{\EulerpolyE{n+1}@{0}}{n+1} = \frac{4(2^{n+2}-1)}{(n+1)(n+2)}\BernoullinumberB{n+2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-2\frac{\EulerpolyE{n+1}@{0}}{n+1} = \frac{4(2^{n+2}-1)}{(n+1)(n+2)}\BernoullinumberB{n+2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- 2*(euler(n + 1, 0))/(n + 1) = (4*((2)^(n + 2)- 1))/((n + 1)*(n + 2))*bernoulli(n + 2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- 2*Divide[EulerE[n + 1, 0],n + 1] == Divide[4*((2)^(n + 2)- 1),(n + 1)*(n + 2)]*BernoulliB[n + 2]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/24.13.E9 24.13.E9] || <math qid="Q7543">\int_{0}^{1/2}\EulerpolyE{2n}@{t}\diff{t} = -\frac{\EulerpolyE{2n+1}@{0}}{2n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1/2}\EulerpolyE{2n}@{t}\diff{t} = -\frac{\EulerpolyE{2n+1}@{0}}{2n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(euler(2*n, t), t = 0..1/2) = -(euler(2*n + 1, 0))/(2*n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[EulerE[2*n, t], {t, 0, 1/2}, GenerateConditions->None] == -Divide[EulerE[2*n + 1, 0],2*n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/24.13.E9 24.13.E9] || <math qid="Q7543">-\frac{\EulerpolyE{2n+1}@{0}}{2n+1} = \frac{2(2^{2n+2}-1)\BernoullinumberB{2n+2}}{(2n+1)(2n+2)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\frac{\EulerpolyE{2n+1}@{0}}{2n+1} = \frac{2(2^{2n+2}-1)\BernoullinumberB{2n+2}}{(2n+1)(2n+2)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>-(euler(2*n + 1, 0))/(2*n + 1) = (2*((2)^(2*n + 2)- 1)*bernoulli(2*n + 2))/((2*n + 1)*(2*n + 2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>-Divide[EulerE[2*n + 1, 0],2*n + 1] == Divide[2*((2)^(2*n + 2)- 1)*BernoulliB[2*n + 2],(2*n + 1)*(2*n + 2)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/24.13.E10 24.13.E10] || <math qid="Q7544">\int_{0}^{1/2}\EulerpolyE{2n-1}@{t}\diff{t} = \frac{\EulernumberE{2n}}{n2^{2n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1/2}\EulerpolyE{2n-1}@{t}\diff{t} = \frac{\EulernumberE{2n}}{n2^{2n+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(euler(2*n - 1, t), t = 0..1/2) = (euler(2*n))/(n*(2)^(2*n + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[EulerE[2*n - 1, t], {t, 0, 1/2}, GenerateConditions->None] == Divide[EulerE[2*n],n*(2)^(2*n + 1)]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/24.13.E11 24.13.E11] || <math qid="Q7545">\int_{0}^{1}\EulerpolyE{n}@{t}\EulerpolyE{m}@{t}\diff{t} = (-1)^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}\BernoullinumberB{m+n+2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}\EulerpolyE{n}@{t}\EulerpolyE{m}@{t}\diff{t} = (-1)^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}\BernoullinumberB{m+n+2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(euler(n, t)*euler(m, t), t = 0..1) = (- 1)^(n)* 4*(((2)^(m + n + 2)- 1)*factorial(m)*factorial(n))/(factorial(m + n + 2))*bernoulli(m + n + 2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[EulerE[n, t]*EulerE[m, t], {t, 0, 1}, GenerateConditions->None] == (- 1)^(n)* 4*Divide[((2)^(m + n + 2)- 1)*(m)!*(n)!,(m + n + 2)!]*BernoulliB[m + n + 2]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|}
</div>