反三角函数积分表

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−1，asin或arcsine表示。

反正弦

• \int \arcsin \frac{x}{c} \ dx = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2}

• \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}

• \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}

• \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)

反正切

• \int \arctan \frac{x}{c} \ dx = x \arctan \frac{x}{c} - \frac{c}{2} \ln(x^2 + c^2)

• \int x \arctan \frac{x}{c} \ dx = \frac{c^2 + x^2}{2} \arctan \frac{x}{c} - \frac{c x}{2}

• \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}

• \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

反正割

• \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|

• \int x \arcsec \frac{x}{c} \ dx = \frac{1}{2} \left( x^2 \arcsec \frac{x}{c} - c \sqrt{x^2 - c^2 } \right)

• \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\}

反余切

• \int \arccot \frac{x}{c} \ dx = x \arccot \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2)

• \int x \arccot \frac{x}{c} \ dx = \frac{c^2 + x^2}{2} \arccot \frac{x}{c} + \frac{c x}{2}

• \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)

• \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