Bernoulli and Euler Polynomials - 24.13 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
24.13.E2 ∫ x x + 1 B n ⁡ ( t ) ⁢ d t = x n superscript subscript 𝑥 𝑥 1 Bernoulli-polynomial-B 𝑛 𝑡 𝑡 superscript 𝑥 𝑛 {\displaystyle{\displaystyle\int_{x}^{x+1}B_{n}\left(t\right)\mathrm{d}t=x^{n}}}
\int_{x}^{x+1}\BernoullipolyB{n}@{t}\diff{t} = x^{n}		 

int(bernoulli(n, t), t = x..x + 1) = (x)^(n)

Integrate[BernoulliB[n, t], {t, x, x + 1}, GenerateConditions->None] == (x)^(n)
Failure Failure Successful [Tested: 3] Successful [Tested: 9]
24.13.E3 ∫ x x + ( 1 / 2 ) B n ⁡ ( t ) ⁢ d t = E n ⁡ ( 2 ⁢ x ) 2 n + 1 superscript subscript 𝑥 𝑥 1 2 Bernoulli-polynomial-B 𝑛 𝑡 𝑡 Euler-polynomial-E 𝑛 2 𝑥 superscript 2 𝑛 1 {\displaystyle{\displaystyle\int_{x}^{x+(1/2)}B_{n}\left(t\right)\mathrm{d}t=% \frac{E_{n}\left(2x\right)}{2^{n+1}}}}
\int_{x}^{x+(1/2)}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulerpolyE{n}@{2x}}{2^{n+1}}		 

int(bernoulli(n, t), t = x..x +(1/2)) = (euler(n, 2*x))/((2)^(n + 1))

Integrate[BernoulliB[n, t], {t, x, x +(1/2)}, GenerateConditions->None] == Divide[EulerE[n, 2*x],(2)^(n + 1)]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.13.E4 ∫ 0 1 / 2 B n ⁡ ( t ) ⁢ d t = 1 - 2 n + 1 2 n ⁢ B n + 1 n + 1 superscript subscript 0 1 2 Bernoulli-polynomial-B 𝑛 𝑡 𝑡 1 superscript 2 𝑛 1 superscript 2 𝑛 Bernoulli-number-B 𝑛 1 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1/2}B_{n}\left(t\right)\mathrm{d}t=\frac% {1-2^{n+1}}{2^{n}}\frac{B_{n+1}}{n+1}}}
\int_{0}^{1/2}\BernoullipolyB{n}@{t}\diff{t} = \frac{1-2^{n+1}}{2^{n}}\frac{\BernoullinumberB{n+1}}{n+1}		 

int(bernoulli(n, t), t = 0..1/2) = (1 - (2)^(n + 1))/((2)^(n))*(bernoulli(n + 1))/(n + 1)

Integrate[BernoulliB[n, t], {t, 0, 1/2}, GenerateConditions->None] == Divide[1 - (2)^(n + 1),(2)^(n)]*Divide[BernoulliB[n + 1],n + 1]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.13.E5 ∫ 1 / 4 3 / 4 B n ⁡ ( t ) ⁢ d t = E n 2 2 ⁢ n + 1 superscript subscript 1 4 3 4 Bernoulli-polynomial-B 𝑛 𝑡 𝑡 Euler-number-E 𝑛 superscript 2 2 𝑛 1 {\displaystyle{\displaystyle\int_{1/4}^{3/4}B_{n}\left(t\right)\mathrm{d}t=% \frac{E_{n}}{2^{2n+1}}}}
\int_{1/4}^{3/4}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulernumberE{n}}{2^{2n+1}}		 

int(bernoulli(n, t), t = 1/4..3/4) = (euler(n))/((2)^(2*n + 1))

Integrate[BernoulliB[n, t], {t, 1/4, 3/4}, GenerateConditions->None] == Divide[EulerE[n],(2)^(2*n + 1)]
Missing Macro Error Failure - Successful [Tested: 3]
24.13.E6 ∫ 0 1 B n ⁡ ( t ) ⁢ B m ⁡ ( t ) ⁢ d t = ( - 1 ) n - 1 ⁢ m ! ⁢ n ! ( m + n ) ! ⁢ B m + n superscript subscript 0 1 Bernoulli-polynomial-B 𝑛 𝑡 Bernoulli-polynomial-B 𝑚 𝑡 𝑡 superscript 1 𝑛 1 𝑚 𝑛 𝑚 𝑛 Bernoulli-number-B 𝑚 𝑛 {\displaystyle{\displaystyle\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)% \mathrm{d}t=\frac{(-1)^{n-1}m!n!}{(m+n)!}B_{m+n}}}
\int_{0}^{1}\BernoullipolyB{n}@{t}\BernoullipolyB{m}@{t}\diff{t} = \frac{(-1)^{n-1}m!n!}{(m+n)!}\BernoullinumberB{m+n}		 

int(bernoulli(n, t)*bernoulli(m, t), t = 0..1) = ((- 1)^(n - 1)* factorial(m)*factorial(n))/(factorial(m + n))*bernoulli(m + n)

Integrate[BernoulliB[n, t]*BernoulliB[m, t], {t, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n - 1)* (m)!*(n)!,(m + n)!]*BernoulliB[m + n]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.13.E8 ∫ 0 1 E n ⁡ ( t ) ⁢ d t = - 2 ⁢ E n + 1 ⁡ ( 0 ) n + 1 superscript subscript 0 1 Euler-polynomial-E 𝑛 𝑡 𝑡 2 Euler-polynomial-E 𝑛 1 0 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1}E_{n}\left(t\right)\mathrm{d}t=-2\frac% {E_{n+1}\left(0\right)}{n+1}}}
\int_{0}^{1}\EulerpolyE{n}@{t}\diff{t} = -2\frac{\EulerpolyE{n+1}@{0}}{n+1}		 

int(euler(n, t), t = 0..1) = - 2*(euler(n + 1, 0))/(n + 1)

Integrate[EulerE[n, t], {t, 0, 1}, GenerateConditions->None] == - 2*Divide[EulerE[n + 1, 0],n + 1]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.13.E8 - 2 ⁢ E n + 1 ⁡ ( 0 ) n + 1 = 4 ⁢ ( 2 n + 2 - 1 ) ( n + 1 ) ⁢ ( n + 2 ) ⁢ B n + 2 2 Euler-polynomial-E 𝑛 1 0 𝑛 1 4 superscript 2 𝑛 2 1 𝑛 1 𝑛 2 Bernoulli-number-B 𝑛 2 {\displaystyle{\displaystyle-2\frac{E_{n+1}\left(0\right)}{n+1}=\frac{4(2^{n+2% }-1)}{(n+1)(n+2)}B_{n+2}}}
-2\frac{\EulerpolyE{n+1}@{0}}{n+1} = \frac{4(2^{n+2}-1)}{(n+1)(n+2)}\BernoullinumberB{n+2}		 

- 2*(euler(n + 1, 0))/(n + 1) = (4*((2)^(n + 2)- 1))/((n + 1)*(n + 2))*bernoulli(n + 2)

- 2*Divide[EulerE[n + 1, 0],n + 1] == Divide[4*((2)^(n + 2)- 1),(n + 1)*(n + 2)]*BernoulliB[n + 2]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.13.E9 ∫ 0 1 / 2 E 2 ⁢ n ⁡ ( t ) ⁢ d t = - E 2 ⁢ n + 1 ⁡ ( 0 ) 2 ⁢ n + 1 superscript subscript 0 1 2 Euler-polynomial-E 2 𝑛 𝑡 𝑡 Euler-polynomial-E 2 𝑛 1 0 2 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1/2}E_{2n}\left(t\right)\mathrm{d}t=-% \frac{E_{2n+1}\left(0\right)}{2n+1}}}
\int_{0}^{1/2}\EulerpolyE{2n}@{t}\diff{t} = -\frac{\EulerpolyE{2n+1}@{0}}{2n+1}		 

int(euler(2*n, t), t = 0..1/2) = -(euler(2*n + 1, 0))/(2*n + 1)

Integrate[EulerE[2*n, t], {t, 0, 1/2}, GenerateConditions->None] == -Divide[EulerE[2*n + 1, 0],2*n + 1]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.13.E9 - E 2 ⁢ n + 1 ⁡ ( 0 ) 2 ⁢ n + 1 = 2 ⁢ ( 2 2 ⁢ n + 2 - 1 ) ⁢ B 2 ⁢ n + 2 ( 2 ⁢ n + 1 ) ⁢ ( 2 ⁢ n + 2 ) Euler-polynomial-E 2 𝑛 1 0 2 𝑛 1 2 superscript 2 2 𝑛 2 1 Bernoulli-number-B 2 𝑛 2 2 𝑛 1 2 𝑛 2 {\displaystyle{\displaystyle-\frac{E_{2n+1}\left(0\right)}{2n+1}=\frac{2(2^{2n% +2}-1)B_{2n+2}}{(2n+1)(2n+2)}}}
-\frac{\EulerpolyE{2n+1}@{0}}{2n+1} = \frac{2(2^{2n+2}-1)\BernoullinumberB{2n+2}}{(2n+1)(2n+2)}		 

-(euler(2*n + 1, 0))/(2*n + 1) = (2*((2)^(2*n + 2)- 1)*bernoulli(2*n + 2))/((2*n + 1)*(2*n + 2))

-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)]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.13.E10 ∫ 0 1 / 2 E 2 ⁢ n - 1 ⁡ ( t ) ⁢ d t = E 2 ⁢ n n ⁢ 2 2 ⁢ n + 1 superscript subscript 0 1 2 Euler-polynomial-E 2 𝑛 1 𝑡 𝑡 Euler-number-E 2 𝑛 𝑛 superscript 2 2 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1/2}E_{2n-1}\left(t\right)\mathrm{d}t=% \frac{E_{2n}}{n2^{2n+1}}}}
\int_{0}^{1/2}\EulerpolyE{2n-1}@{t}\diff{t} = \frac{\EulernumberE{2n}}{n2^{2n+1}}		 

int(euler(2*n - 1, t), t = 0..1/2) = (euler(2*n))/(n*(2)^(2*n + 1))

Integrate[EulerE[2*n - 1, t], {t, 0, 1/2}, GenerateConditions->None] == Divide[EulerE[2*n],n*(2)^(2*n + 1)]
Missing Macro Error Failure - Skipped - Because timed out
24.13.E11 ∫ 0 1 E n ⁡ ( t ) ⁢ E m ⁡ ( t ) ⁢ d t = ( - 1 ) n ⁢ 4 ⁢ ( 2 m + n + 2 - 1 ) ⁢ m ! ⁢ n ! ( m + n + 2 ) ! ⁢ B m + n + 2 superscript subscript 0 1 Euler-polynomial-E 𝑛 𝑡 Euler-polynomial-E 𝑚 𝑡 𝑡 superscript 1 𝑛 4 superscript 2 𝑚 𝑛 2 1 𝑚 𝑛 𝑚 𝑛 2 Bernoulli-number-B 𝑚 𝑛 2 {\displaystyle{\displaystyle\int_{0}^{1}E_{n}\left(t\right)E_{m}\left(t\right)% \mathrm{d}t=(-1)^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}B_{m+n+2}}}
\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}		 

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)

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]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
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