查看“︁反三角函数积分表”︁的源代码

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{{三角学}}
以下是部份[[反三角函數]]的[[積分]]表。(书写时省略了不定积分结果中都含有的任意常数Cn)

同一個反三角函數亦有多種的表達方式，其中有三種是最常用的。如sine的反函數可以以''sin<sup>−1</sup>''，''asin''或arcsine表示。


== 反正弦 ==
* <math>\int \arcsin \frac{x}{c} \  dx = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2}</math>

* <math>\int x \arcsin \frac{x}{c} \  dx = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arcsin \frac{x}{c} + \frac{x}{4} \sqrt{c^2 - x^2}</math>

* <math>\int x^2 \arcsin \frac{x}{c} \  dx = \frac{x^3}{3} \arcsin \frac{x}{c} + \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}</math>

* <math>\int x^n \arcsin x \  dx = \frac{1}{n + 1} \left( x^{n + 1} \arcsin x + \frac{x^n \sqrt{1 - x^2} - n x^{n - 1} \arcsin x}{n + 1} + \frac{n(n-1)}{n+1} \int x^{n - 2} \arcsin x \  dx \right)</math>

== 反正切 ==
* <math>\int \arctan \frac{x}{c} \  dx = x \arctan \frac{x}{c} - \frac{c}{2} \ln(x^2 + c^2)</math>

* <math>\int x \arctan \frac{x}{c} \  dx = \frac{c^2 + x^2}{2} \arctan \frac{x}{c} - \frac{c x}{2}</math>

* <math>\int x^2 \arctan \frac{x}{c} \  dx = \frac{x^3}{3} \arctan \frac{x}{c} - \frac{c x^2}{6} + \frac{c^3}{6} \ln{c^2 + x^2}</math>

* <math>\int x^n \arctan \frac{x}{c} \  dx = \frac{x^{n + 1}}{n + 1} \arctan \frac{x}{c} - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \  dx, \quad n \neq 1</math>

== 反正割 ==
* <math>\int \arcsec \frac{x}{c} \  dx = x \arcsec \frac{x}{c} - c \sgn(x)\ln \left| x + \sqrt{x^2 - c ^2} \right|
                               = x \arcsec \frac{x}{c} + c \sgn(x)\ln \left| x - \sqrt{x^2 - c ^2} \right|</math>

* <math>\int x \arcsec \frac{x}{c}  \ dx = \frac{1}{2} \left( x^2 \arcsec \frac{x}{c} - c \sqrt{x^2 - c^2 } \right)</math>

* <math>\int x^n \arcsec x \  dx = \frac{1}{n + 1} \left\{ x^{n + 1} \arcsec x - \frac{1}{n} \left[ x^{n - 1} \sqrt{x^2 - 1} + (1 - n) \left( x^{n - 1} \arcsec x + (1 - n) \int x^{n - 2} \arcsec x \  dx \right) \right] \right\}</math>

== 反余切 ==
* <math>\int \arccot \frac{x}{c} \  dx = x \arccot \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2)</math>

* <math>\int x \arccot \frac{x}{c} \  dx = \frac{c^2 + x^2}{2} \arccot \frac{x}{c} + \frac{c x}{2}</math>

* <math>\int x^2 \arccot \frac{x}{c} \  dx = \frac{x^3}{3} \arccot \frac{x}{c} + \frac{c x^2}{6} - \frac{c^3}{6} \ln(c^2 + x^2)</math>

* <math>\int x^n \arccot \frac{x}{c} \  dx = \frac{x^{n + 1}}{n+1} \arccot \frac{x}{c} + \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx, \quad n \neq 1</math>

{{Lists of integrals}}

[[Category:积分表]]