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# Fungrim entry: 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))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC