# §24.13(i) Bernoulli Polynomials

 24.13.1 $\displaystyle\int\mathop{B_{n}\/}\nolimits\!\left(t\right)dt$ $\displaystyle=\frac{\mathop{B_{n+1}\/}\nolimits\!\left(t\right)}{n+1}+\text{% const.},$ 24.13.2 $\displaystyle\int_{x}^{x+1}\mathop{B_{n}\/}\nolimits\!\left(t\right)dt$ $\displaystyle=x^{n},$ $n=1,2,\dots$, 24.13.3 $\displaystyle\int_{x}^{x+(1/2)}\mathop{B_{n}\/}\nolimits\!\left(t\right)dt$ $\displaystyle=\frac{\mathop{E_{n}\/}\nolimits\!\left(2x\right)}{2^{n+1}},$ 24.13.4 $\displaystyle\int_{0}^{1/2}\mathop{B_{n}\/}\nolimits\!\left(t\right)dt$ $\displaystyle=\frac{1-2^{n+1}}{2^{n}}\frac{\mathop{B_{n+1}\/}\nolimits}{n+1},$ 24.13.5 $\displaystyle\int_{1/4}^{3/4}\mathop{B_{n}\/}\nolimits\!\left(t\right)dt$ $\displaystyle=\frac{\mathop{E_{n}\/}\nolimits}{2^{2n+1}}.$

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

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

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

# §24.13(ii) Euler Polynomials

 24.13.7 $\int\mathop{E_{n}\/}\nolimits\!\left(t\right)dt=\frac{\mathop{E_{n+1}\/}% \nolimits\!\left(t\right)}{n+1}+\text{const.},$
 24.13.8 $\int_{0}^{1}\mathop{E_{n}\/}\nolimits\!\left(t\right)dt=-2\frac{\mathop{E_{n+1% }\/}\nolimits\!\left(0\right)}{n+1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}\mathop{B_{% n+2}\/}\nolimits,$
 24.13.9 $\int_{0}^{1/2}\mathop{E_{2n}\/}\nolimits\!\left(t\right)dt=-\frac{\mathop{E_{2% n+1}\/}\nolimits\!\left(0\right)}{2n+1}=\frac{2(2^{2n+2}-1)\mathop{B_{2n+2}\/}% \nolimits}{(2n+1)(2n+2)},$
 24.13.10 $\int_{0}^{1/2}\mathop{E_{2n-1}\/}\nolimits\!\left(t\right)dt=\frac{\mathop{E_{% 2n}\/}\nolimits}{n2^{2n+1}},$ $n=1,2,\dots$.

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

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

# §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).