Definite integrals

Table of contents: Powers - Exponential functions - Sophomore's dream

Powers

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

Exponential functions

$\int_{z}^{\infty} {e}^{-a x + b} \, dx = \frac{{e}^{b - a z}}{a}$
$\int_{z}^{\infty} {x}^{c} {e}^{-a x + b} \, dx = \frac{{e}^{b}}{{a}^{c + 1}} \Gamma\!\left(c + 1, a z\right)$
$\int_{z}^{\infty} {e}^{-a {x}^{2} + b} \, dx = \frac{{e}^{b}}{2} \sqrt{\frac{\pi}{a}} \operatorname{erfc}\!\left(\sqrt{a} z\right)$
$\int_{z}^{\infty} {e}^{-a {x}^{c} + b} \, dx = \frac{{e}^{b}}{c {a}^{1 / c}} \Gamma\!\left(\frac{1}{c}, a {z}^{c}\right)$

Sophomore's dream

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

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

2020-08-27 09:56:25.682319 UTC