Engineering Handbook/Calculus/Integration/inverse trigonometric functions

Arcsine function integration formulas

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

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

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

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

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

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

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

Arccosine function integration formulas

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

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

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

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

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

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

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

Arctangent function integration formulas

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

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

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

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

Arccotangent function integration formulas

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

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

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

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

Arcsecant function integration formulas

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

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

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

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

Arccosecant function integration formulas

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

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

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

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