逆三角関数
導関数を積分し一点で値を固定すると逆三角関数の定積分としての表現が得られる:
arcsin  x = ∫ 0 x d z 1 − z 2 , | x | ≤ 1 arccos  x = ∫ x 1 d z 1 − z 2 , | x | ≤ 1 arctan  x = ∫ 0 x d z z 2 + 1 , arccot  x = ∫ x ∞ d z z 2 + 1 , arcsec  x = ∫ 1 x d z z z 2 − 1 , x ≥ 1 arcsec  x = π + ∫ x − 1 d z z z 2 − 1 , x ≤ − 1 arccsc  x = ∫ x ∞ d z z z 2 − 1 , x ≥ 1 arccsc  x = ∫ − ∞ x d z z z 2 − 1 , x ≤ − 1 {\displaystyle {\begin{aligned}\arcsin x&=\int _{0}^{x}{\frac {dz}{\sqrt {1-z^{2}}}},\qquad |x|\leq 1\\\arccos x&=\int _{x}^{1}{\frac {dz}{\sqrt {1-z^{2}}}},\qquad |x|\leq 1\\\arctan x&=\int _{0}^{x}{\frac {dz}{z^{2}+1}},\\\operatorname {arccot} x&=\int _{x}^{\infty }{\frac {dz}{z^{2}+1}},\\\operatorname {arcsec} x&=\int _{1}^{x}{\frac {dz}{z{\sqrt {z^{2}-1}}}},\qquad x\geq 1\\\operatorname {arcsec} x&=\pi +\int _{x}^{-1}{\frac {dz}{z{\sqrt {z^{2}-1}}}},\qquad x\leq -1\\\operatorname {arccsc} x&=\int _{x}^{\infty }{\frac {dz}{z{\sqrt {z^{2}-1}}}},\qquad x\geq 1\\\operatorname {arccsc} x&=\int _{-\infty }^{x}{\frac {dz}{z{\sqrt {z^{2}-1}}}},\qquad x\leq -1\end{aligned}}}