Engineering Handbook/Calculus/Integration/inverse trigonometric functions

Arcsine function integration formulas

 * $$\int\arcsin(a\,x)\,dx=

x\arcsin(a\,x)+ \frac{\sqrt{1-a^2\,x^2}}{a}+C$$


 * $$\int x\arcsin(a\,x)\,dx=

\frac{x^2\arcsin(a\,x)}{2}- \frac{\arcsin(a\,x)}{4\,a^2}+ \frac{x\sqrt{1-a^2\,x^2}}{4\,a}+C$$


 * $$\int x^2\arcsin(a\,x)\,dx=

\frac{x^3\arcsin(a\,x)}{3}+ \frac{\left(a^2\,x^2+2\right)\sqrt{1-a^2\,x^2}}{9\,a^3}+C$$


 * $$\int x^m\arcsin(a\,x)\,dx=

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


 * $$\int\arcsin(a\,x)^2\,dx=

-2\,x+x\arcsin(a\,x)^2+ \frac{2\sqrt{1-a^2\,x^2}\arcsin(a\,x)}{a}+C$$


 * $$\int\arcsin(a\,x)^n\,dx=

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


 * $$\int\arcsin(a\,x)^n\,dx=

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

Arccosine function integration formulas

 * $$\int\arccos(a\,x)\,dx=

x\arccos(a\,x)- \frac{\sqrt{1-a^2\,x^2}}{a}+C$$


 * $$\int x\arccos(a\,x)\,dx=

\frac{x^2\arccos(a\,x)}{2}- \frac{\arccos(a\,x)}{4\,a^2}- \frac{x\sqrt{1-a^2\,x^2}}{4\,a}+C$$


 * $$\int x^2\arccos(a\,x)\,dx=

\frac{x^3\arccos(a\,x)}{3}- \frac{\left(a^2\,x^2+2\right)\sqrt{1-a^2\,x^2}}{9\,a^3}+C$$


 * $$\int x^m\arccos(a\,x)\,dx=

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


 * $$\int\arccos(a\,x)^2\,dx=

-2\,x+x\arccos(a\,x)^2- \frac{2\sqrt{1-a^2\,x^2}\arccos(a\,x)}{a}+C$$


 * $$\int\arccos(a\,x)^n\,dx=

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


 * $$\int\arccos(a\,x)^n\,dx=

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

Arctangent function integration formulas

 * $$\int\arctan(a\,x)\,dx=

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


 * $$\int x\arctan(a\,x)\,dx=

\frac{x^2\arctan(a\,x)}{2}+ \frac{\arctan(a\,x)}{2\,a^2}-\frac{x}{2\,a}+C$$


 * $$\int x^2\arctan(a\,x)\,dx=

\frac{x^3\arctan(a\,x)}{3}+ \frac{\ln\left(a^2\,x^2+1\right)}{6\,a^3}-\frac{x^2}{6\,a}+C$$


 * $$\int x^m\arctan(a\,x)\,dx=

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

Arccotangent function integration formulas

 * $$\int\arccot(a\,x)\,dx=

x\arccot(a\,x)+ \frac{\ln\left(a^2\,x^2+1\right)}{2\,a}+C$$


 * $$\int x\arccot(a\,x)\,dx=

\frac{x^2\arccot(a\,x)}{2}+ \frac{\arccot(a\,x)}{2\,a^2}+\frac{x}{2\,a}+C$$


 * $$\int x^2\arccot(a\,x)\,dx=

\frac{x^3\arccot(a\,x)}{3}- \frac{\ln\left(a^2\,x^2+1\right)}{6\,a^3}+\frac{x^2}{6\,a}+C$$


 * $$\int x^m\arccot(a\,x)\,dx=

\frac{x^{m+1}\arccot(a\,x)}{m+1}+ \frac{a}{m+1}\int \frac{x^{m+1}}{a^2\,x^2+1}\,dx\quad(m\ne-1)$$

Arcsecant function integration formulas

 * $$\int\arcsec(a\,x)\,dx=

x\arcsec(a\,x)- \frac{1}{a}\,\operatorname{arctanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}+C$$


 * $$\int x\arcsec(a\,x)\,dx=

\frac{x^2\arcsec(a\,x)}{2}- \frac{x}{2\,a}\sqrt{1-\frac{1}{a^2\,x^2}}+C$$


 * $$\int x^2\arcsec(a\,x)\,dx=

\frac{x^3\arcsec(a\,x)}{3}\,-\, \frac{1}{6\,a^3}\,\operatorname{arctanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}\,-\, \frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2\,x^2}}\,+\,C$$


 * $$\int x^m\arcsec(a\,x)\,dx=

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

Arccosecant function integration formulas

 * $$\int\arccsc(a\,x)\,dx=

x\arccsc(a\,x)+ \frac{1}{a}\,\operatorname{arctanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}+C$$


 * $$\int x\arccsc(a\,x)\,dx=

\frac{x^2\arccsc(a\,x)}{2}+ \frac{x}{2\,a}\sqrt{1-\frac{1}{a^2\,x^2}}+C$$


 * $$\int x^2\arccsc(a\,x)\,dx=

\frac{x^3\arccsc(a\,x)}{3}\,+\, \frac{1}{6\,a^3}\,\operatorname{arctanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}\,+\, \frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2\,x^2}}\,+\,C$$


 * $$\int x^m\arccsc(a\,x)\,dx=

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