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)

三角学

历史三角函数 (反三角函数) 广义三角函数
参考
恒等式精确值三角表单位圆
定理
正弦餘弦正切餘切勾股定理
微积分
三角换元法积分 (反三角函数) 微分

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

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

$\int x\arcsin {\frac {x}{c}}\ dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arcsin {\frac {x}{c}}+{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}$

$\int x^{2}\arcsin {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{c}}+{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}$

$\int x^{n}\arcsin x\ dx={\frac {1}{n+1}}\left(x^{n+1}\arcsin x+{\frac {x^{n}{\sqrt {1-x^{2}}}-nx^{n-1}\arcsin x}{n+1}}+{\frac {n(n-1)}{n+1}}\int x^{n-2}\arcsin x\ dx\right)$

反正切

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

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

$\int x^{2}\arctan {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\arctan {\frac {x}{c}}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}$

$\int x^{n}\arctan {\frac {x}{c}}\ dx={\frac {x^{n+1}}{n+1}}\arctan {\frac {x}{c}}-{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx,\quad n\neq 1$

反正割

$\int \operatorname {arcsec} {\frac {x}{a}}\ dx=x\operatorname {arcsec} {\frac {x}{a}}-a\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|=x\operatorname {arcsec} {\frac {x}{a}}+a\operatorname {sgn}(x)\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|$

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

$\int x^{n}\operatorname {arcsec} x\ dx={\frac {1}{n+1}}\left\{x^{n+1}\operatorname {arcsec} x-{\frac {1}{n}}\left[x^{n-1}{\sqrt {x^{2}-1}}+(1-n)\left(x^{n-1}\operatorname {arcsec} x+(1-n)\int x^{n-2}\operatorname {arcsec} x\ dx\right)\right]\right\}$

反余切

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

$\int x\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx}{2}}$

$\int x^{2}\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})$

$\int x^{n}\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx,\quad n\neq 1$

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