From Wikipedia, the free encyclopedia
The following is a list of
integrals (
antiderivative functions) of
irrational functions. For a complete list of integral functions, see
lists of integrals. Throughout this article the
constant of integration is omitted for brevity.
Integrals involving r = √a2 + x2


























Integrals involving s = √x2 − a2
Assume x2 > a2 (for x2 < a2, see next section):



where the positive value of
is to be taken.














![{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054a5959ce5e03cf279c1b29dff2ba014ac6dcde)
![{\displaystyle \int {\frac {dx}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86843311de7fc72bc01f87742445f7c4b88899e9)
![{\displaystyle \int {\frac {dx}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\frac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32b3a8d7f9040840f5d1de3467129edff0d80b)

![{\displaystyle \int {\frac {x^{2}\,dx}{s^{7}}}={\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a98057cf3f3d6b7025114445c972bb6b7b7af9d)
![{\displaystyle \int {\frac {x^{2}\,dx}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cce4b87e7a47ce42042803038139f830afd5d37)
Integrals involving u = √a2 − x2








Integrals involving R = √ax2 + bx + c
Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.






















Integrals involving S = √ax + b







- Abramowitz, Milton; Stegun, Irene A., eds. (1972).
"Chapter 3".
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover.
-
Gradshteyn, Izrail Solomonovich;
Ryzhik, Iosif Moiseevich;
Geronimus, Yuri Veniaminovich;
Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel;
Moll, Victor Hugo (eds.).
Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.).
Academic Press, Inc.
ISBN
978-0-12-384933-5.
LCCN
2014010276. (Several previous editions as well.)
- Peirce, Benjamin Osgood (1929) [1899]. "Chapter 3". A Short Table of Integrals (3rd revised ed.). Boston: Ginn and Co. pp. 16–30.