# §24.13 Integrals

## §24.13(i) Bernoulli Polynomials

 24.13.1 $\displaystyle\int B_{n}\left(t\right)\,\mathrm{d}t$ $\displaystyle=\frac{B_{n+1}\left(t\right)}{n+1}+\text{const.},$ ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$: 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 and Ch.24 24.13.2 $\displaystyle\int_{x}^{x+1}B_{n}\left(t\right)\,\mathrm{d}t$ $\displaystyle=x^{n},$ $n=1,2,\dots$, 24.13.3 $\displaystyle\int_{x}^{x+(1/2)}B_{n}\left(t\right)\,\mathrm{d}t$ $\displaystyle=\frac{E_{n}\left(2x\right)}{2^{n+1}},$ 24.13.4 $\displaystyle\int_{0}^{1/2}B_{n}\left(t\right)\,\mathrm{d}t$ $\displaystyle=\frac{1-2^{n+1}}{2^{n}}\frac{B_{n+1}}{n+1},$ 24.13.5 $\displaystyle\int_{1/4}^{3/4}B_{n}\left(t\right)\,\mathrm{d}t$ $\displaystyle=\frac{E_{n}}{2^{2n+1}}.$

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

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

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

## §24.13(ii) Euler Polynomials

 24.13.7 $\int E_{n}\left(t\right)\,\mathrm{d}t=\frac{E_{n+1}\left(t\right)}{n+1}+\text{% const.},$ ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$: 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 and Ch.24
 24.13.8 $\int_{0}^{1}E_{n}\left(t\right)\,\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+% 1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},$
 24.13.9 $\int_{0}^{1/2}E_{2n}\left(t\right)\,\mathrm{d}t=-\frac{E_{2n+1}\left(0\right)}% {2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},$
 24.13.10 $\int_{0}^{1/2}E_{2n-1}\left(t\right)\,\mathrm{d}t=\frac{E_{2n}}{n2^{2n+1}},$ $n=1,2,\dots$.

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

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

## §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).