逆三角関数の原始関数の一覧
∫ x m arccos  a x d x = x m + 1 arccos  a x m + 1 + a m + 1 ∫ x m + 1 1 − a 2 x 2 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\arccos ax\,dx={\frac {x^{m+1}\arccos ax}{m+1}}\,+\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}x^{2}}}}\,dx\quad (m\neq -1)}
∫ ( arccos  a x ) 2 d x = − 2 x + x ( arccos  a x ) 2 − 2 1 − a 2 x 2 arccos  a x a + C {\displaystyle \int (\arccos ax)^{2}\,dx=-2\,x+x\,(\arccos ax)^{2}-{\frac {2{\sqrt {1-a^{2}x^{2}}}\arccos ax}{a}}+C}
∫ x m arcsin  a x d x = x m + 1 arcsin  a x m + 1 − a m + 1 ∫ x m + 1 1 − a 2 x 2 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\arcsin ax\,dx={\frac {x^{m+1}\arcsin ax}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}x^{2}}}}\,dx\quad (m\neq -1)}
∫ ( arcsin  a x ) 2 d x = − 2 x + x ( arcsin  a x ) 2 + 2 1 − a 2 x 2 arcsin  a x a + C {\displaystyle \int (\arcsin ax)^{2}\,dx=-2\,x+x\,(\arcsin ax)^{2}+{\frac {2{\sqrt {1-a^{2}x^{2}}}\arcsin ax}{a}}+C}
∫ arccos  x d x = x arccos  x − 1 − x 2 + C {\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}+C}
∫ arccos  a x d x = x arccos  a x − 1 − a 2 x 2 a + C {\displaystyle \int \arccos ax\,dx=x\arccos ax-{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}
∫ x arccos  a x d x = x 2 arccos  a x 2 − arccos  a x 4 a 2 − x 1 − a 2 x 2 4 a + C {\displaystyle \int x\arccos ax\,dx={\frac {x^{2}\arccos ax}{2}}-{\frac {\arccos ax}{4\,a^{2}}}-{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4\,a}}+C}
∫ x 2 arccos  a x d x = x 3 arccos  a x 3 − ( a 2 x 2 + 2 ) 1 − a 2 x 2 9 a 3 + C {\displaystyle \int x^{2}\arccos ax\,dx={\frac {x^{3}\arccos ax}{3}}-{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}}}}{9\,a^{3}}}+C}