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。 ∫ arcsin ⁡ x c   d x = x arcsin ⁡ x c + c 2 − x 2 {\displaystyle \int \arcsin {\frac {x}{c}}\ dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}} ∫ x arcsin ⁡ x c   d x = ( x 2 2 − c 2 4 ) arcsin ⁡ x c + x 4 c 2 − x 2 {\displaystyle \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}}}} ∫ x 2 arcsin ⁡ x c   d x = x 3 3 arcsin ⁡ x c + x 2 + 2 c 2 9 c 2 − x 2 {\displaystyle \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}}}} ∫ x n arcsin ⁡ x   d x = 1 n + 1 ( x n + 1 arcsin ⁡ x + x n 1 − x 2 − n x n − 1 arcsin ⁡ x n + 1 + n ( n − 1 ) n + 1 ∫ x n − 2 arcsin ⁡ x   d x ) {\displaystyle \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)} ∫ arctan ⁡ x c   d x = x arctan ⁡ x c − c 2 ln ⁡ ( x 2 + c 2 ) {\displaystyle \int \arctan {\frac {x}{c}}\ dx=x\arctan {\frac {x}{c}}-{\frac {c}{2}}\ln(x^{2}+c^{2})} ∫ x arctan ⁡ x c   d x = c 2 + x 2 2 arctan ⁡ x c − c x 2 {\displaystyle \int x\arctan {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\arctan {\frac {x}{c}}-{\frac {cx}{2}}} ∫ x 2 arctan ⁡ x c   d x = x 3 3 arctan ⁡ x c − c x 2 6 + c 3 6 ln ⁡ c 2 + x 2 {\displaystyle \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}}} ∫ x n arctan ⁡ x c   d x = x n + 1 n + 1 arctan ⁡ x c − c n + 1 ∫ x n + 1 c 2 + x 2   d x , n ≠ 1 {\displaystyle \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} ∫ arcsec ⁡ x a   d x = x arcsec ⁡ x a − a sgn ⁡ ( x ) ln ⁡ | x + x 2 − a 2 | = x arcsec ⁡ x a + a sgn ⁡ ( x ) ln ⁡ | x − x 2 − a 2 | {\displaystyle \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|} ∫ x arcsec ⁡ x   d x = 1 2 ( x 2 arcsec ⁡ x − x 2 − 1 ) {\displaystyle \int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)} ∫ x n arcsec ⁡ x   d x = 1 n + 1 { x n + 1 arcsec ⁡ x − 1 n [ x n − 1 x 2 − 1 + ( 1 − n ) ( x n − 1 arcsec ⁡ x + ( 1 − n ) ∫ x n − 2 arcsec ⁡ x   d x ) ] } {\displaystyle \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\}} ∫ arccot ⁡ x c   d x = x arccot ⁡ x c + c 2 ln ⁡ ( c 2 + x 2 ) {\displaystyle \int \operatorname {arccot} {\frac {x}{c}}\ dx=x\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})} ∫ x arccot ⁡ x c   d x = c 2 + x 2 2 arccot ⁡ x c + c x 2 {\displaystyle \int x\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx}{2}}} ∫ x 2 arccot ⁡ x c   d x = x 3 3 arccot ⁡ x c + c x 2 6 − c 3 6 ln ⁡ ( c 2 + x 2 ) {\displaystyle \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})} ∫ x n arccot ⁡ x c   d x = x n + 1 n + 1 arccot ⁡ x c + c n + 1 ∫ x n + 1 c 2 + x 2   d x , n ≠ 1 {\displaystyle \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}