Списак интеграла инверзних тригонометријских функција

Овде је дат списак интеграла инверзних тригонометријских функција.

Аркуссинус

${\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}$

${\displaystyle \int \arcsin {\frac {x}{a}}\ dx=x\arcsin {\frac {x}{a}}+{\sqrt {a^{2}-x^{2}}}+C}$

${\displaystyle \int x\arcsin {\frac {x}{a}}\ dx=\left({\frac {x^{2}}{2}}\frac {a^{2}}{4}}\right)\arcsin {\frac {x}{a}}+{\frac {x}{4}}{\sqrt {a^{2x^{2}}}+C}$

${\displaystyle \int x^{2}\arcsin {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{a}}+{\frac {x^{2}+2a^{2}}{9}}{\sqrt {a^{2}-x^{2}}}+C}$

${\displaystyle \int x^{n}\arcsin x\ dx={\frac {1}{n+1}}\left(x^{n+1}\arcsin x+{\frac {x^{n}{\sqrt {1-x^{2}}}-nx^{n-1}\arcsin x}{n-1}}+n\int x^{n-2}\arcsin x\ dx\right)}$

${\displaystyle \int \cos ^{n}x\arcsin x\ dx=\left(x^{n^{2}+1}\arccos x+{\frac {x^{n}{\sqrt {1-x^{4}}}-nx^{n^{2}-1}\arccos x}{n^{2}-1}}+n\int x^{n^{2}-2}\arccos x\ dx\right)}$

Аркускосинус

${\displaystyle \int \arccos x\,dx=x\arccos x\sqrt {1-x^{2}}}+C}$

${\displaystyle \int \arccos {\frac {x}{a}}\ dx=x\arccos {\frac {x}{a}}\sqrt {a^{2x^{2}}}+C}$

${\displaystyle \int x\arccos {\frac {x}{a}}\ dx=\left({\frac {x^{2}}{2}}\frac {a^{2}}{4}}\right)\arccos {\frac {x}{a}{\frac {x}{4}}{\sqrt {a^{2}-x^{2}}}+C}$

${\displaystyle \int x^{2}\arccos {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\arccos {\frac {x}{a}}\frac {x^{2}+2a^{2}}{9}}{\sqrt {a^{2x^{2}}}+C}$

Аркустангенс

${\displaystyle \int \arctan x\,dx=x\arctan x\frac {1}{2}}\ln(1+x^{2})+C}$

${\displaystyle \int \arctan {\big (}{\frac {x}{a}}{\big )}dx=x\arctan {\big (}{\frac {x}{a}}{\big ){\frac {a}{2}}\ln(1+{\frac {x^{2}}{a^{2}}})+C}$

${\displaystyle \int x\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {(a^{2}+x^{2})\arctan {\big (}{\frac {x}{a}}{\big )}-ax}{2}}+C}$

${\displaystyle \int x^{2}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{3}}{3}}\arctan {\big (}{\frac {x}{a}}{\big )}\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({a^{2}+x^{2}})+C}$

${\displaystyle \int x^{n}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{n+1}}{n+1}}\arctan {\big (}{\frac {x}{a}}{\big )}\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ dx,\quad n\neq -1}$

Аркускотангенс

${\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}$

${\displaystyle \int \operatorname {arccsc} {\frac {x}{a}}\ dx=x\operatorname {arccsc} {\frac {x}{a}}+{a}\ln {({\frac {x}{a}}({\sqrt {1 x + x x 2 − 1 x 2 | + C {\displaystyle \int \operatorname {arcsec} x\,dx=x\operatorname {arcsec} x-\ln \left|x+x{\sqrt {{x^{21} \over x^{2}}}\right|+C}$

${\displaystyle \int \operatorname {arcsec} {\frac {x}{a}}\ dx=x\operatorname {arcsec} {\frac {x}{a}}+{\frac {x}{a|x|}}\ln \left|x\pm {\sqrt {x^{21}}\right|+C}$

${\displaystyle \int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x\sqrt {x^{21}}\right)+C}$

${\displaystyle \int x^{n}\operatorname {arcsec} x\ dx={\frac {1}{n+1}}\left(x^{n+1}\operatorname {arcsec} x\frac {1}{n}}\left[x^{n-1}{\sqrt {x^{21}}+[1-n]\left(x^{n-1}\operatorname {arcsec} x+(1-n)\int x^{n-2}\operatorname {arcsec} x\ dx\right)\right]\right)}$

Аркускотангенс

${\displaystyle \int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+{\frac {1}{2}}\ln(1+x^{2})+C}$

${\displaystyle \int \operatorname {arccot} {\frac {x}{a}}\ dx=x\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{2}}\ln(a^{2}+x^{2})+C}$

${\displaystyle \int x\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {a^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax}{2}}+C}$

${\displaystyle \int x^{2}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax^{2}}{6}}\frac {a^{3}}{6}}\ln(a^{2}+x^{2})+C}$

${\displaystyle \int x^{n}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\ dx,\quad n\neq -1}$

Види још

• Интеграл
• Инверзне тригонометријске функције
• Списак интеграла тригонометријских функција

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.  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.  978-2-88124-097-3.. 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. .
• Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002.  978-1-58488-291-6.. (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)