Definite integrals

463077

\int_{z}^{\infty} \frac{1}{{\left(a x + b\right)}^{c}} \, dx = \frac{1}{a \left(c - 1\right) {\left(a z + b\right)}^{c - 1}}

Assumptions:

a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, b \in \mathbb{R} \,\mathbin{\operatorname{and}}\, c \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \gt 0 \,\mathbin{\operatorname{and}}\, a z + b \gt 0 \,\mathbin{\operatorname{and}}\, c \gt 1

TeX:

\int_{z}^{\infty} \frac{1}{{\left(a x + b\right)}^{c}} \, dx = \frac{1}{a \left(c - 1\right) {\left(a z + b\right)}^{c - 1}}

a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, b \in \mathbb{R} \,\mathbin{\operatorname{and}}\, c \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \gt 0 \,\mathbin{\operatorname{and}}\, a z + b \gt 0 \,\mathbin{\operatorname{and}}\, c \gt 1

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("463077"),
    Formula(Equal(Integral(Div(1, Pow(Add(Mul(a, x), b), c)), Tuple(x, z, Infinity)), Div(1, Mul(Mul(a, Sub(c, 1)), Pow(Add(Mul(a, z), b), Sub(c, 1)))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, RR), Element(b, RR), Element(c, RR), Element(z, RR), Greater(a, 0), Greater(Add(Mul(a, z), b), 0), Greater(c, 1))))

02e3d2

\int_{z}^{\infty} {e}^{-a x + b} \, dx = \frac{{e}^{b - a z}}{a}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \gt 0

TeX:

\int_{z}^{\infty} {e}^{-a x + b} \, dx = \frac{{e}^{b - a z}}{a}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("02e3d2"),
    Formula(Equal(Integral(Exp(Add(Neg(Mul(a, x)), b)), Tuple(x, z, Infinity)), Div(Exp(Sub(b, Mul(a, z))), a))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Greater(Re(a), 0))))

9a06fb

\int_{z}^{\infty} {x}^{c} {e}^{-a x + b} \, dx = \frac{{e}^{b}}{{a}^{c + 1}} \Gamma\!\left(c + 1, a z\right)

Assumptions:

a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, b \in \mathbb{R} \,\mathbin{\operatorname{and}}\, c \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \gt 0 \,\mathbin{\operatorname{and}}\, c \gt 0 \,\mathbin{\operatorname{and}}\, z \gt 0

TeX:

\int_{z}^{\infty} {x}^{c} {e}^{-a x + b} \, dx = \frac{{e}^{b}}{{a}^{c + 1}} \Gamma\!\left(c + 1, a z\right)

a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, b \in \mathbb{R} \,\mathbin{\operatorname{and}}\, c \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \gt 0 \,\mathbin{\operatorname{and}}\, c \gt 0 \,\mathbin{\operatorname{and}}\, z \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("9a06fb"),
    Formula(Equal(Integral(Mul(Pow(x, c), Exp(Add(Neg(Mul(a, x)), b))), Tuple(x, z, Infinity)), Mul(Div(Exp(b), Pow(a, Add(c, 1))), UpperGamma(Add(c, 1), Mul(a, z))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, RR), Element(b, RR), Element(c, RR), Element(z, RR), Greater(a, 0), Greater(c, 0), Greater(z, 0))))

f8de2e

\int_{z}^{\infty} {e}^{-a {x}^{2} + b} \, dx = \frac{{e}^{b}}{2} \sqrt{\frac{\pi}{a}} \operatorname{erfc}\!\left(\sqrt{a} z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \gt 0

TeX:

\int_{z}^{\infty} {e}^{-a {x}^{2} + b} \, dx = \frac{{e}^{b}}{2} \sqrt{\frac{\pi}{a}} \operatorname{erfc}\!\left(\sqrt{a} z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Erfc`	$\operatorname{erfc}\!\left(z\right)$	Complementary error function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("f8de2e"),
    Formula(Equal(Integral(Exp(Add(Neg(Mul(a, Pow(x, 2))), b)), Tuple(x, z, Infinity)), Mul(Mul(Div(Exp(b), 2), Sqrt(Div(ConstPi, a))), Erfc(Mul(Sqrt(a), z))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Greater(Re(a), 0))))

16a1f4

\int_{z}^{\infty} {e}^{-a {x}^{c} + b} \, dx = \frac{{e}^{b}}{c {a}^{1 / c}} \Gamma\!\left(\frac{1}{c}, a {z}^{c}\right)

Assumptions:

a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, b \in \mathbb{R} \,\mathbin{\operatorname{and}}\, c \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \gt 0 \,\mathbin{\operatorname{and}}\, c \gt 0 \,\mathbin{\operatorname{and}}\, z \gt 0

TeX:

\int_{z}^{\infty} {e}^{-a {x}^{c} + b} \, dx = \frac{{e}^{b}}{c {a}^{1 / c}} \Gamma\!\left(\frac{1}{c}, a {z}^{c}\right)

a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, b \in \mathbb{R} \,\mathbin{\operatorname{and}}\, c \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \gt 0 \,\mathbin{\operatorname{and}}\, c \gt 0 \,\mathbin{\operatorname{and}}\, z \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("16a1f4"),
    Formula(Equal(Integral(Exp(Add(Neg(Mul(a, Pow(x, c))), b)), Tuple(x, z, Infinity)), Mul(Div(Exp(b), Mul(c, Pow(a, Div(1, c)))), UpperGamma(Div(1, c), Mul(a, Pow(z, c)))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, RR), Element(b, RR), Element(c, RR), Element(z, RR), Greater(a, 0), Greater(c, 0), Greater(z, 0))))

b77faf

\int_{0}^{1} {x}^{-x} \, dx = \sum_{n=1}^{\infty} {n}^{-n}

TeX:

\int_{0}^{1} {x}^{-x} \, dx = \sum_{n=1}^{\infty} {n}^{-n}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("b77faf"),
    Formula(Equal(Integral(Pow(x, Neg(x)), Tuple(x, 0, 1)), Sum(Pow(n, Neg(n)), Tuple(n, 1, Infinity)))))

66fefb

\int_{0}^{1} {x}^{x} \, dx = \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} {n}^{-n}

TeX:

\int_{0}^{1} {x}^{x} \, dx = \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} {n}^{-n}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("66fefb"),
    Formula(Equal(Integral(Pow(x, x), Tuple(x, 0, 1)), Sum(Mul(Pow(-1, Add(n, 1)), Pow(n, Neg(n))), Tuple(n, 1, Infinity)))))

Powers

Exponential functions

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