Списак интеграла инверзних тригонометријских функција

Овде је дат списак интеграла инверзних тригонометријских функција.

Аркуссинус

$\int \arcsin xdx=x\arcsin x+\sqrt{1-x^2}+C$

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

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

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

$\int x^n\arcsin xdx=\frac{1}{n+1}(x^{n+1}\arcsin x+\frac{x^n\sqrt{1-x^2}-n{x}^{n-1}\arcsin x}{n-1}+n\int x^{n-2}\arcsin xdx)$

$\int {\cos }^{n}x\arcsin xdx=(x^{n^2+1}\arccos x+\frac{x^n\sqrt{1-x^4}-n{x}^{n^2-1}\arccos x}{n^2-1}+n\int x^{n^2-2}\arccos xdx)$

Аркускосинус

$\int \arccos xdx=x\arccos x-\sqrt{1-x^2}+C$

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

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

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

Аркустангенс

$\int \arctan xdx=x\arctan x-\frac{1}{2}\ln (1+x^2)+C$

$\int \arctan \left(\frac{x}{a}\right)dx=x\arctan \left(\frac{x}{a}\right)-\frac{a}{2}\ln (1+\frac{x^2}{a^2})+C$

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

$\int x^2\arctan \left(\frac{x}{a}\right)dx=\frac{x^3}{3}\arctan \left(\frac{x}{a}\right)-\frac{a{x}^2}{6}+\frac{a^3}{6}\ln (a^2+x^2)+C$

$\int x^n\arctan \left(\frac{x}{a}\right)dx=\frac{x^{n+1}}{n+1}\arctan \left(\frac{x}{a}\right)-\frac{a}{n+1}\int \frac{x^{n+1}}{a^2+x^2}dx,n\ne -1$

Аркускотангенс

$\int \mathrm{arccsc}xdx=x\mathrm{arccsc}x+\ln |x+x\sqrt{\frac{x^2-1}{x^2}}|+C$

$\int \mathrm{arccsc}\frac{x}{a}dx=x\mathrm{arccsc}\frac{x}{a}+a\ln (\frac{x}{a}(\sqrt{1-\frac{a^2}{x^2}}+1))+C$

$\int x\mathrm{arccsc}\frac{x}{a}dx=\frac{x^2}{2}\mathrm{arccsc}\frac{x}{a}+\frac{ax}{2}\sqrt{1-\frac{a^2}{x^2}}+C$

Аркуссеканс

$\int \mathrm{arcsec}xdx=x\mathrm{arcsec}x-\ln |x+x\sqrt{\frac{x^2-1}{x^2}}|+C$

$\int \mathrm{arcsec}\frac{x}{a}dx=x\mathrm{arcsec}\frac{x}{a}+\frac{x}{a|x|}\ln |x\pm \sqrt{x^2-1}|+C$

$\int x\mathrm{arcsec}xdx=\frac{1}{2}(x^2\mathrm{arcsec}x-\sqrt{x^2-1})+C$

$\int x^n\mathrm{arcsec}xdx=\frac{1}{n+1}(x^{n+1}\mathrm{arcsec}x-\frac{1}{n}[x^{n-1}\sqrt{x^2-1}+[1-n](x^{n-1}\mathrm{arcsec}x+(1-n)\int x^{n-2}\mathrm{arcsec}xdx)])$

Аркускотангенс

$\int \mathrm{arccot}xdx=x\mathrm{arccot}x+\frac{1}{2}\ln (1+x^2)+C$

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

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

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

$\int x^n\mathrm{arccot}\frac{x}{a}dx=\frac{x^{n+1}}{n+1}\mathrm{arccot}\frac{x}{a}+\frac{a}{n+1}\int \frac{x^{n+1}}{a^2+x^2}dx,n\ne -1$

Види још

• Интеграл
• Инверзне тригонометријске функције
• Списак интеграла тригонометријских функција

Литература

Шаблон:Литература-{

• Шаблон:Cite book
• I.S. Gradshteyn (И. С. Градштейн), I.M. Ryzhik (И. М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Шаблон:Cite book, seventh edition. . 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–Шаблон:Page. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
• Yu.A. Brychkov (Ю. А. Брычков), Шаблон:Cite book. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition. .
• Шаблон:Cite book, . Chapman & Hall/CRC Press, . (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)

}-Шаблон:Литература крај