International Standard Serial Number

2014-01-24T14:19:14Z

223.196.158.55: /* External links */

{{Trigonometry}}
The following is a list of [[indefinite integral]]s ([[antiderivative]]s) of expressions involving the [[inverse trigonometric function]]s. For a complete list of integral formulas, see [[lists of integrals]].

* The inverse trigonometric functions are also known as the "arc functions".
* ''C'' is used for the arbitrary [[constant of integration]] that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.
* There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as ''sin<sup>−1</sup>'', ''asin'', or, as is used on this page, ''arcsin''.
* For each inverse trigonometric integration formula below there is a corresponding formula in the [[list of integrals of inverse hyperbolic functions]].

== Arcsine function integration formulas ==

:<math>\int\arcsin(x)\,dx=
x\arcsin(x)+
{\sqrt{1-x^2}}+C</math>

:<math>\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</math>

:<math>\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</math>

:<math>\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)</math>

:<math>\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</math>

:<math>\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</math>

:<math>\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)</math>

== Arccosine function integration formulas ==
:<math>\int\arccos(x)\,dx=
x\arccos(x)-
{\sqrt{1-x^2}}+C</math>

:<math>\int\arccos(a\,x)\,dx=
x\arccos(a\,x)-
\frac{\sqrt{1-a^2\,x^2}}{a}+C</math>

:<math>\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</math>

:<math>\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</math>

:<math>\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)</math>

:<math>\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</math>

:<math>\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</math>

:<math>\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)</math>

== Arctangent function integration formulas ==
:<math>\int\arctan(x)\,dx=
x\arctan(x)-
\frac{\ln\left(x^2+1\right)}{2}+C</math>

:<math>\int\arctan(a\,x)\,dx=
x\arctan(a\,x)-
\frac{\ln\left(a^2\,x^2+1\right)}{2\,a}+C</math>

:<math>\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</math>

:<math>\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</math>

:<math>\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)</math>

== Arccotangent function integration formulas ==

:<math>\int\arccot(x)\,dx=
x\arccot(x)+
\frac{\ln\left(x^2+1\right)}{2}+C</math>

:<math>\int\arccot(a\,x)\,dx=
x\arccot(a\,x)+
\frac{\ln\left(a^2\,x^2+1\right)}{2\,a}+C</math>

:<math>\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</math>

:<math>\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</math>

:<math>\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)</math>

== Arcsecant function integration formulas ==

:<math>\int\arcsec(x)\,dx=
x\arcsec(x)-\operatorname{arctan}\,\sqrt{1-\frac{1}{x^2}}+C</math>

:<math>\int\arcsec(a\,x)\,dx=
x\arcsec(a\,x)-
\frac{1}{a}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}+C</math>

:<math>\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</math>

:<math>\int x^2\arcsec(a\,x)\,dx=
\frac{x^3\arcsec(a\,x)}{3}\,-\,
\frac{1}{6\,a^3}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}\,-\,
\frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2\,x^2}}\,+\,C</math>

:<math>\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)</math>

== Arccosecant function integration formulas ==
:<math>\int\arccsc(x)\,dx=
x\arccos(x)-
\sqrt{1-{x^2}}+C</math>

:<math>\int\arccsc(a\,x)\,dx=
x\arccsc(a\,x)+
\frac{1}{a}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}+C</math>

:<math>\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</math>

:<math>\int x^2\arccsc(a\,x)\,dx=
\frac{x^3\arccsc(a\,x)}{3}\,+\,
\frac{1}{6\,a^3}\,\operatorname{artanh}\,\sqrt{1-\frac{1}{a^2\,x^2}}\,+\,
\frac{x^2}{6\,a}\sqrt{1-\frac{1}{a^2\,x^2}}\,+\,C</math>

:<math>\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)</math>

{{Lists of integrals}}

[[Category:Integrals|Arc functions]]
[[Category:Mathematics-related lists|Integrals of arc functions]]

Schmidt number

2012-09-01T06:22:09Z

223.196.151.158:

Let me first start by introducing myself. My title is Boyd Butts even though it is not the name on my birth certificate. Her spouse and her reside in Puerto Rico but she will have to transfer 1 day or an additional. He used to be unemployed but now he is a meter reader. Body developing is what my family and I enjoy.<br><br>Here is my web site [http://www.1A-pornotube.com/user/WCrompton home std test]

formulasearchengine - User contributions [en]

International Standard Serial Number

Schmidt number