Fungrim entry: 7adfd6

\int_{a}^{b} B_{n}\!\left(t\right) \, dt = \frac{B_{n + 1}\!\left(b\right) - B_{n + 1}\!\left(a\right)}{n + 1}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\int_{a}^{b} B_{n}\!\left(t\right) \, dt = \frac{B_{n + 1}\!\left(b\right) - B_{n + 1}\!\left(a\right)}{n + 1}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7adfd6"),
    Formula(Equal(Integral(BernoulliPolynomial(n, t), For(t, a, b)), Div(Sub(BernoulliPolynomial(Add(n, 1), b), BernoulliPolynomial(Add(n, 1), a)), Add(n, 1)))),
    Variables(n, a, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC))))

Topics using this entry

Bernoulli numbers and polynomials