# §24.13 Integrals

## §24.13(i) Bernoulli Polynomials

 24.13.1 $\displaystyle\int\mathop{B_{n}\/}\nolimits\!\left(t\right)\mathrm{d}t$ $\displaystyle=\frac{\mathop{B_{n+1}\/}\nolimits\!\left(t\right)}{n+1}+\text{% const.},$ Symbols: $\mathop{B_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Bernoulli polynomials, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $n$: integer and $t$: real or complex A&S Ref: 23.1.11 Referenced by: §24.13(i) Permalink: http://dlmf.nist.gov/24.13.E1 Encodings: TeX, pMML, png See also: Annotations for 24.13(i) 24.13.2 $\displaystyle\int_{x}^{x+1}\mathop{B_{n}\/}\nolimits\!\left(t\right)\mathrm{d}t$ $\displaystyle=x^{n},$ $n=1,2,\dots$, 24.13.3 $\displaystyle\int_{x}^{x+(1/2)}\mathop{B_{n}\/}\nolimits\!\left(t\right)% \mathrm{d}t$ $\displaystyle=\frac{\mathop{E_{n}\/}\nolimits\!\left(2x\right)}{2^{n+1}},$ 24.13.4 $\displaystyle\int_{0}^{1/2}\mathop{B_{n}\/}\nolimits\!\left(t\right)\mathrm{d}t$ $\displaystyle=\frac{1-2^{n+1}}{2^{n}}\frac{B_{n+1}}{n+1},$ 24.13.5 $\displaystyle\int_{1/4}^{3/4}\mathop{B_{n}\/}\nolimits\!\left(t\right)\mathrm{% d}t$ $\displaystyle=\frac{E_{n}}{2^{2n+1}}.$

For $m,n=1,2,\ldots$,

 24.13.6 $\int_{0}^{1}\mathop{B_{n}\/}\nolimits\!\left(t\right)\mathop{B_{m}\/}\nolimits% \!\left(t\right)\mathrm{d}t=\frac{(-1)^{n-1}m!n!}{(m+n)!}B_{m+n}.$

For integrals of the form $\int_{0}^{x}\mathop{B_{n}\/}\nolimits\!\left(t\right)\mathop{B_{m}\/}\nolimits% \!\left(t\right)\mathrm{d}t$ and $\int_{0}^{x}\mathop{B_{n}\/}\nolimits\!\left(t\right)\mathop{B_{m}\/}\nolimits% \!\left(t\right)\mathop{B_{k}\/}\nolimits\!\left(t\right)\mathrm{d}t$ see Agoh and Dilcher (2011).

## §24.13(ii) Euler Polynomials

 24.13.7 $\int\mathop{E_{n}\/}\nolimits\!\left(t\right)\mathrm{d}t=\frac{\mathop{E_{n+1}% \/}\nolimits\!\left(t\right)}{n+1}+\text{const.},$ Symbols: $\mathop{E_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Euler polynomials, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $n$: integer and $t$: real or complex A&S Ref: 23.1.11 Referenced by: §24.13(ii) Permalink: http://dlmf.nist.gov/24.13.E7 Encodings: TeX, pMML, png See also: Annotations for 24.13(ii)
 24.13.8 $\int_{0}^{1}\mathop{E_{n}\/}\nolimits\!\left(t\right)\mathrm{d}t=-2\frac{% \mathop{E_{n+1}\/}\nolimits\!\left(0\right)}{n+1}=\frac{4(2^{n+2}-1)}{(n+1)(n+% 2)}B_{n+2},$
 24.13.9 $\int_{0}^{1/2}\mathop{E_{2n}\/}\nolimits\!\left(t\right)\mathrm{d}t=-\frac{% \mathop{E_{2n+1}\/}\nolimits\!\left(0\right)}{2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2% }}{(2n+1)(2n+2)},$
 24.13.10 $\int_{0}^{1/2}\mathop{E_{2n-1}\/}\nolimits\!\left(t\right)\mathrm{d}t=\frac{E_% {2n}}{n2^{2n+1}},$ $n=1,2,\dots$.

For $m,n=1,2,\ldots$,

 24.13.11 $\int_{0}^{1}\mathop{E_{n}\/}\nolimits\!\left(t\right)\mathop{E_{m}\/}\nolimits% \!\left(t\right)\mathrm{d}t=(-1)^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}B_{m+n+% 2}.$

## §24.13(iii) Compendia

For Laplace and inverse Laplace transforms see Prudnikov et al. (1992a, §§3.28.1–3.28.2) and Prudnikov et al. (1992b, §§3.26.1–3.26.2). For other integrals see Prudnikov et al. (1990, pp. 55–57).