Список на интеграли на инверзни тригонометриски функции

Списоков е попис на интегралите (антидеривации на функции) на рационалните функции за интегранди кои содржат инверзни тригонометриски функции (познати и како „аркус функции“). За целосен список на интеграли на функции, да се погледа таблични интеграли.

Постојат три вообичаени начини за обележување на инверзните тригонометриски функции. На пример функцијата arcus sinus би можела да се запише како „sin⁻¹“, „asin'“, или „arcsin“ како што е користено во оваа статија.

Arcus sinus

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

$\int \arcsin (ax)dx=x\arcsin (ax)+\frac{\sqrt{1-a^2{x}^2}}{a}+C$

$\int x\arcsin (ax)dx=\frac{x^2\arcsin (ax)}{2}-\frac{\arcsin (ax)}{4a^2}+\frac{x\sqrt{1-a^2{x}^2}}{4a}+C$

$\int x^2\arcsin (ax)dx=\frac{x^3\arcsin (ax)}{3}+\frac{(a^2{x}^2+2)\sqrt{1-a^2{x}^2}}{9a^3}+C$

$\int x^m\arcsin (ax)dx=\frac{x^{m+1}\arcsin (ax)}{m+1}-\frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2{x}^2}}dx(m\ne -1)$

$\int \arcsin (ax{)}^{2}dx=-2x+x\arcsin (ax{)}^2+\frac{2\sqrt{1-a^2{x}^2}\arcsin (ax)}{a}+C$

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

$\int \arcsin (ax{)}^{n}dx=\frac{x\arcsin (ax{)}^{n+2}}{(n+1)(n+2)}+\frac{\sqrt{1-a^2{x}^2}\arcsin (ax{)}^{n+1}}{a(n+1)}-\frac{1}{(n+1)(n+2)}\int \arcsin (ax{)}^{n+2}dx(n\ne -1,-2)$

Arcus cosinus

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

$\int \arccos (ax)dx=x\arccos (ax)-\frac{\sqrt{1-a^2{x}^2}}{a}+C$

$\int x\arccos (ax)dx=\frac{x^2\arccos (ax)}{2}-\frac{\arccos (ax)}{4a^2}-\frac{x\sqrt{1-a^2{x}^2}}{4a}+C$

$\int x^2\arccos (ax)dx=\frac{x^3\arccos (ax)}{3}-\frac{(a^2{x}^2+2)\sqrt{1-a^2{x}^2}}{9a^3}+C$

$\int x^m\arccos (ax)dx=\frac{x^{m+1}\arccos (ax)}{m+1}+\frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2{x}^2}}dx(m\ne -1)$

$\int \arccos (ax{)}^{2}dx=-2x+x\arccos (ax{)}^2-\frac{2\sqrt{1-a^2{x}^2}\arccos (ax)}{a}+C$

$\int \arccos (ax{)}^{n}dx=x\arccos (ax{)}^n-\frac{n\sqrt{1-a^2{x}^2}\arccos (ax{)}^{n-1}}{a}-n(n-1)\int \arccos (ax{)}^{n-2}dx$

$\int \arccos (ax{)}^{n}dx=\frac{x\arccos (ax{)}^{n+2}}{(n+1)(n+2)}-\frac{\sqrt{1-a^2{x}^2}\arccos (ax{)}^{n+1}}{a(n+1)}-\frac{1}{(n+1)(n+2)}\int \arccos (ax{)}^{n+2}dx(n\ne -1,-2)$

Arcus tangens

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

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

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

$\int x^2\arctan (ax)dx=\frac{x^3\arctan (ax)}{3}+\frac{\ln (a^2{x}^2+1)}{6a^3}-\frac{x^2}{6a}+C$

$\int x^m\arctan (ax)dx=\frac{x^{m+1}\arctan (ax)}{m+1}-\frac{a}{m+1}\int \frac{x^{m+1}}{a^2{x}^2+1}dx(m\ne -1)$

Arcus cotangens

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

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

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

$\int x^2\mathrm{arccot}(ax)dx=\frac{x^3\mathrm{arccot}(ax)}{3}-\frac{\ln (a^2{x}^2+1)}{6a^3}+\frac{x^2}{6a}+C$

$\int x^m\mathrm{arccot}(ax)dx=\frac{x^{m+1}\mathrm{arccot}(ax)}{m+1}+\frac{a}{m+1}\int \frac{x^{m+1}}{a^2{x}^2+1}dx(m\ne -1)$

Arcus secans

$\int \mathrm{arcsec}(x)dx=x\mathrm{arcsec}(x)-\mathrm{arcosh}|x|+C$

$\int \mathrm{arcsec}(ax)dx=x\mathrm{arcsec}(ax)-\frac{1}{a}\mathrm{arcosh}|ax|+C$

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

$\int x^2\mathrm{arcsec}(ax)dx=\frac{x^3\mathrm{arcsec}(ax)}{3}-\frac{\mathrm{arcosh}|ax|}{6a^3}-\frac{x^2}{6a}\sqrt{1-\frac{1}{a^2{x}^2}}+C$

$\int x^m\mathrm{arcsec}(ax)dx=\frac{x^{m+1}\mathrm{arcsec}(ax)}{m+1}-\frac{1}{a(m+1)}\int \frac{x^{m-1}}{\sqrt{1-\frac{1}{a^2{x}^2}}}dx(m\ne -1)$

Arcus cosecans

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

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

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

$\int x^2\mathrm{arccsc}(ax)dx=\frac{x^3\mathrm{arccsc}(ax)}{3}+\frac{1}{6a^3}\mathrm{artanh}\sqrt{1-\frac{1}{a^2{x}^2}}+\frac{x^2}{6a}\sqrt{1-\frac{1}{a^2{x}^2}}+C$

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