Списак интеграла ирационалних функција

Списак интеграла ирационалних функција:

Интеграли са a² - x²

$\int\sqrt{a^2-x^2}\;dx = \frac{1}{2}\left(x\sqrt{a^2-x^2}+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int x\sqrt{a^2-x^2}\;dx = -\frac{1}{3}\sqrt{(a^2-x^2)^3} \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int\frac{\sqrt{a^2-x^2}\;dx}{x} = \sqrt{a^2-x^2}-a\ln\left|\frac{a+\sqrt{a^2+x^2}}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int\frac{dx}{\sqrt{a^2-x^2}} = \arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int\frac{x^2\;dx}{\sqrt{a^2-x^2}} = -\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}$

Интеграли са x² + a²

$\int\sqrt{x^2+a^2}\;dx = \frac{1}{2}\left(x\sqrt{x^2+a^2}+a^2\,\mathrm{arcsinh}\frac{x}{a}\right)$

$\int x\sqrt{x^2+a^2}\;dx=\frac13\sqrt{(x^2+a^2)^3}$

$\int\frac{\sqrt{x^2+a^2}\;dx}{x} = \sqrt{x^2+a^2}-a\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right|$

$\int\frac{dx}{\sqrt{x^2+a^2}} = \mathrm{arcsinh}\frac{x}{a} = \ln\left|x+\sqrt{x^2+a^2}\right|$

$\int\frac{x\,dx}{\sqrt{x^2+a^2}} = \sqrt{x^2+a^2}$

$\int\frac{x^2\;dx}{\sqrt{x^2+a^2}} = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\,\mathrm{arcsinh}\frac{x}{a} = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln\left|x+\sqrt{x^2+a^2}\right|$

$\int\frac{dx}{x\sqrt{x^2+a^2}} = -\frac{1}{a}\,\mathrm{arcsinh}\frac{a}{x} = -\frac{1}{a}\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right|$

Интеграли са x² - a²

$\int\sqrt{x^2-a^2}\;dx = \frac{1}{2}\left(x\sqrt{x^2-a^2}-\sgn x\,\mathrm{arccosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(za }|x|\ge|a|\mbox{)}$

$\int x\sqrt{x^2-a^2}\;dx = \frac{1}{3}\sqrt{(x^2-a^2)^3} \qquad\mbox{(za }|x|\ge|a|\mbox{)}$

$\int\frac{\sqrt{x^2-a^2}\;dx}{x} = \sqrt{x^2-a^2} - a\arccos\frac{a}{x} \qquad\mbox{(za }|x|\ge|a|\mbox{)}$

$\int\frac{dx}{\sqrt{x^2-a^2}} = \mathrm{arccosh}\frac{x}{a} = \ln\left(|x|+\sqrt{x^2-a^2}\right) \qquad\mbox{(za }|x|>|a|\mbox{)}$

$\int\frac{x\;dx}{\sqrt{x^2-a^2}} = \sqrt{x^2-a^2} \qquad\mbox{(za }|x|>|a|\mbox{)}$

$\int\frac{x^2\,dx}{\sqrt{x^2-a^2}} = \frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\,\mathrm{arccosh}\left|\frac{x}{a}\right| = \frac{1}{2}\left(x\sqrt{x^2-a^2}+a^2\ln\left(|x|+\sqrt{x^2-a^2}\right)\right) \qquad\mbox{(za }|x|>|a|\mbox{)}$

Интеграли са ax² + bx + c

$\int{\sqrt{ax^2+bx+c}\;dx} = \frac{1}{8a^{3/2}} \left (2\sqrt{a} \cdot \left (b + 2ax \right ) \sqrt{c + x \left (b + ax \right )} - \left (b^2 - 4ac \right ) \log{ \left (b + 2x + 2\sqrt{a} \sqrt{ c + x \left (b + ax\right ) } \right ) } \right )$

$\int\frac{dx}{\sqrt{ax^2+bx+c}} = \frac{1}{\sqrt{a}}\ln\left|2\sqrt{a(ax^2+bx+c)}+2ax+b\right| \qquad\mbox{(za }a>0\mbox{)}$

$\int\frac{dx}{\sqrt{ax^2+bx+c}} = \frac{1}{\sqrt{a}}\,\mathrm{arcsinh}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(za }a>0\mbox{, }4ac-b^2>0\mbox{)}$

$\int\frac{dx}{\sqrt{ax^2+bx+c}} = \frac{1}{\sqrt{a}}\ln|2ax+b| \qquad\mbox{(za }a>0\mbox{, }4ac-b^2=0\mbox{)}$

$\int\frac{dx}{\sqrt{ax^2+bx+c}} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad\mbox{(za }a<0\mbox{, }4ac-b^2<0\mbox{)}$

$\int\frac{x\;dx}{\sqrt{ax^2+bx+c}} = \frac{\sqrt{ax^2+bx+c}}{a}-\frac{b}{2a}\int\frac{dx}{\sqrt{ax^2+bx+c}}$

Литература

• Milton Abramowitz and Irene Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
• I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)
• A.P. Prudnikov (А.П. Прудников), Yu.A. Brychkov (Ю.А. Брычков), O.I. Marichev (О.И. Маричев). Integrals and Series. First edition (Russian), volume 1–5, Nauka, 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/CRC Press, 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
• Yu.A. Brychkov (Ю.А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X.
• Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.)
• Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
• Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)