逆双曲線関数の原始関数の一覧

プロジェクト 数学 ポータル 数学 逆双曲線関数の原始関数の一覧（ぎゃくそうきょくせんかんすうのせきぶんほうのいちらん）では、逆双曲線関数の原始関数を一覧形式でまとめた。原始関数の一覧も参照のこと。 以下の数式において、定数a は0ではないものとし、C は積分定数とする。 以下の数式はそれぞれ逆三角関数の原始関数の一覧の数式と対応する。 ∫ arsinh ⁡ ( a x ) d x = x arsinh ⁡ ( a x ) − a 2 x 2 + 1 a + C {\displaystyle \int \operatorname {arsinh} (a\,x)\,dx=x\,\operatorname {arsinh} (a\,x)-{\frac {\sqrt {a^{2}\,x^{2}+1}}{a}}+C} ∫ x arsinh ⁡ ( a x ) d x = x 2 arsinh ⁡ ( a x ) 2 + arsinh ⁡ ( a x ) 4 a 2 − x a 2 x 2 + 1 4 a + C {\displaystyle \int x\,\operatorname {arsinh} (a\,x)dx={\frac {x^{2}\,\operatorname {arsinh} (a\,x)}{2}}+{\frac {\operatorname {arsinh} (a\,x)}{4\,a^{2}}}-{\frac {x{\sqrt {a^{2}\,x^{2}+1}}}{4\,a}}+C} ∫ x 2 arsinh ⁡ ( a x ) d x = x 3 arsinh ⁡ ( a x ) 3 − ( a 2 x 2 − 2 ) a 2 x 2 + 1 9 a 3 + C {\displaystyle \int x^{2}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{3}\,\operatorname {arsinh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}-2\right){\sqrt {a^{2}\,x^{2}+1}}}{9\,a^{3}}}+C} ∫ x m arsinh ⁡ ( a x ) d x = x m + 1 arsinh ⁡ ( a x ) m + 1 − a m + 1 ∫ x m + 1 a 2 x 2 + 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsinh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {a^{2}\,x^{2}+1}}}\,dx\quad (m\neq -1)} ∫ arsinh ⁡ ( a x ) 2 d x = 2 x + x arsinh ⁡ ( a x ) 2 − 2 a 2 x 2 + 1 arsinh ⁡ ( a x ) a + C {\displaystyle \int \operatorname {arsinh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arsinh} (a\,x)^{2}-{\frac {2\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)}{a}}+C} ∫ arsinh ⁡ ( a x ) n d x = x arsinh ⁡ ( a x ) n − n a 2 x 2 + 1 arsinh ⁡ ( a x ) n − 1 a + n ( n − 1 ) ∫ arsinh ⁡ ( a x ) n − 2 d x {\displaystyle \int \operatorname {arsinh} (a\,x)^{n}\,dx=x\,\operatorname {arsinh} (a\,x)^{n}\,-\,{\frac {n\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)^{n-1}}{a}}\,+\,n\,(n-1)\int \operatorname {arsinh} (a\,x)^{n-2}\,dx} ∫ arsinh ⁡ ( a x ) n d x = − x arsinh ⁡ ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a 2 x 2 + 1 arsinh ⁡ ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) ∫ arsinh ⁡ ( a x ) n + 2 d x ( n ≠ − 1 , − 2 ) {\displaystyle \int \operatorname {arsinh} (a\,x)^{n}\,dx=-{\frac {x\,\operatorname {arsinh} (a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)^{n+1}}{a(n+1)}}\,+\,{\frac {1}{(n+1)\,(n+2)}}\int \operatorname {arsinh} (a\,x)^{n+2}\,dx\quad (n\neq -1,-2)} ∫ arcosh ⁡ ( a x ) d x = x arcosh ⁡ ( a x ) − a x + 1 a x − 1 a + C {\displaystyle \int \operatorname {arcosh} (a\,x)\,dx=x\,\operatorname {arcosh} (a\,x)-{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{a}}+C} ∫ x arcosh ⁡ ( a x ) d x = x 2 arcosh ⁡ ( a x ) 2 − arcosh ⁡ ( a x ) 4 a 2 − x a x + 1 a x − 1 4 a + C {\displaystyle \int x\,\operatorname {arcosh} (a\,x)dx={\frac {x^{2}\,\operatorname {arcosh} (a\,x)}{2}}-{\frac {\operatorname {arcosh} (a\,x)}{4\,a^{2}}}-{\frac {x\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{4\,a}}+C} ∫ x 2 arcosh ⁡ ( a x ) d x = x 3 arcosh ⁡ ( a x ) 3 − ( a 2 x 2 + 2 ) a x + 1 a x − 1 9 a 3 + C {\displaystyle \int x^{2}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{3}\,\operatorname {arcosh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}+2\right){\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{9\,a^{3}}}+C} ∫ x m arcosh ⁡ ( a x ) d x = x m + 1 arcosh ⁡ ( a x ) m + 1 − a m + 1 ∫ x m + 1 a x + 1 a x − 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arcosh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}}\,dx\quad (m\neq -1)} ∫ arcosh ⁡ ( a x ) 2 d x = 2 x + x arcosh ⁡ ( a x ) 2 − 2 a x + 1 a x − 1 arcosh ⁡ ( a x ) a + C {\displaystyle \int \operatorname {arcosh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arcosh} (a\,x)^{2}-{\frac {2\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)}{a}}+C} ∫ arcosh ⁡ ( a x ) n d x = x arcosh ⁡ ( a x ) n − n a x + 1 a x − 1 arcosh ⁡ ( a x ) n − 1 a + n ( n − 1 ) ∫ arcosh ⁡ ( a x ) n − 2 d x {\displaystyle \int \operatorname {arcosh} (a\,x)^{n}\,dx=x\,\operatorname {arcosh} (a\,x)^{n}\,-\,{\frac {n\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)^{n-1}}{a}}\,+\,n\,(n-1)\int \operatorname {arcosh} (a\,x)^{n-2}\,dx} ∫ arcosh ⁡ ( a x ) n d x = − x arcosh ⁡ ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a x + 1 a x − 1 arcosh ⁡ ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) ∫ arcosh ⁡ ( a x ) n + 2 d x ( n ≠ − 1 , − 2 ) {\displaystyle \int \operatorname {arcosh} (a\,x)^{n}\,dx=-{\frac {x\,\operatorname {arcosh} (a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)^{n+1}}{a\,(n+1)}}\,+\,{\frac {1}{(n+1)\,(n+2)}}\int \operatorname {arcosh} (a\,x)^{n+2}\,dx\quad (n\neq -1,-2)} ∫ artanh ⁡ ( a x ) d x = x artanh ⁡ ( a x ) + ln ⁡ ( a 2 x 2 − 1 ) 2 a + C {\displaystyle \int \operatorname {artanh} (a\,x)\,dx=x\,\operatorname {artanh} (a\,x)+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{2\,a}}+C} ∫ x artanh ⁡ ( a x ) d x = x 2 artanh ⁡ ( a x ) 2 − artanh ⁡ ( a x ) 2 a 2 + x 2 a + C {\displaystyle \int x\,\operatorname {artanh} (a\,x)dx={\frac {x^{2}\,\operatorname {artanh} (a\,x)}{2}}-{\frac {\operatorname {artanh} (a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C} ∫ x 2 artanh ⁡ ( a x ) d x = x 3 artanh ⁡ ( a x ) 3 + ln ⁡ ( a 2 x 2 − 1 ) 6 a 3 + x 2 6 a + C {\displaystyle \int x^{2}\,\operatorname {artanh} (a\,x)dx={\frac {x^{3}\,\operatorname {artanh} (a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C} ∫ x m artanh ⁡ ( a x ) d x = x m + 1 artanh ⁡ ( a x ) m + 1 + a m + 1 ∫ x m + 1 a 2 x 2 − 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {artanh} (a\,x)dx={\frac {x^{m+1}\operatorname {artanh} (a\,x)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}-1}}\,dx\quad (m\neq -1)} ∫ arcoth ⁡ ( a x ) d x = x arcoth ⁡ ( a x ) + ln ⁡ ( a 2 x 2 − 1 ) 2 a + C {\displaystyle \int \operatorname {arcoth} (a\,x)\,dx=x\,\operatorname {arcoth} (a\,x)+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{2\,a}}+C} ∫ x arcoth ⁡ ( a x ) d x = x 2 arcoth ⁡ ( a x ) 2 − arcoth ⁡ ( a x ) 2 a 2 + x 2 a + C {\displaystyle \int x\,\operatorname {arcoth} (a\,x)dx={\frac {x^{2}\,\operatorname {arcoth} (a\,x)}{2}}-{\frac {\operatorname {arcoth} (a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C} ∫ x 2 arcoth ⁡ ( a x ) d x = x 3 arcoth ⁡ ( a x ) 3 + ln ⁡ ( a 2 x 2 − 1 ) 6 a 3 + x 2 6 a + C {\displaystyle \int x^{2}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{3}\,\operatorname {arcoth} (a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C} ∫ x m arcoth ⁡ ( a x ) d x = x m + 1 arcoth ⁡ ( a x ) m + 1 + a m + 1 ∫ x m + 1 a 2 x 2 − 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{m+1}\operatorname {arcoth} (a\,x)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}-1}}\,dx\quad (m\neq -1)} ∫ arsech ⁡ ( a x ) d x = x arsech ⁡ ( a x ) − 2 a arctan ⁡ 1 − a x 1 + a x + C {\displaystyle \int \operatorname {arsech} (a\,x)\,dx=x\,\operatorname {arsech} (a\,x)-{\frac {2}{a}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}+C} ∫ x arsech ⁡ ( a x ) d x = x 2 arsech ⁡ ( a x ) 2 − ( 1 + a x ) 2 a 2 1 − a x 1 + a x + C {\displaystyle \int x\,\operatorname {arsech} (a\,x)dx={\frac {x^{2}\,\operatorname {arsech} (a\,x)}{2}}-{\frac {(1+a\,x)}{2\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}+C} ∫ x 2 arsech ⁡ ( a x ) d x = x 3 arsech ⁡ ( a x ) 3 − 1 3 a 3 arctan ⁡ 1 − a x 1 + a x − x ( 1 + a x ) 6 a 2 1 − a x 1 + a x + C {\displaystyle \int x^{2}\,\operatorname {arsech} (a\,x)dx={\frac {x^{3}\,\operatorname {arsech} (a\,x)}{3}}\,-\,{\frac {1}{3\,a^{3}}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}\,-\,{\frac {x(1+a\,x)}{6\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}\,+\,C} ∫ x m arsech ⁡ ( a x ) d x = x m + 1 arsech ⁡ ( a x ) m + 1 + 1 m + 1 ∫ x m ( 1 + a x ) 1 − a x 1 + a x d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arsech} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsech} (a\,x)}{m+1}}\,+\,{\frac {1}{m+1}}\int {\frac {x^{m}}{(1+a\,x){\sqrt {\frac {1-a\,x}{1+a\,x}}}}}\,dx\quad (m\neq -1)} ∫ arcsch ⁡ ( a x ) d x = x arcsch ⁡ ( a x ) + 1 a arcoth ⁡ 1 a 2 x 2 + 1 + C {\displaystyle \int \operatorname {arcsch} (a\,x)\,dx=x\,\operatorname {arcsch} (a\,x)+{\frac {1}{a}}\,\operatorname {arcoth} {\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}+C} ∫ x arcsch ⁡ ( a x ) d x = x 2 arcsch ⁡ ( a x ) 2 + x 2 a 1 a 2 x 2 + 1 + C {\displaystyle \int x\,\operatorname {arcsch} (a\,x)dx={\frac {x^{2}\,\operatorname {arcsch} (a\,x)}{2}}+{\frac {x}{2\,a}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}+C} ∫ x 2 arcsch ⁡ ( a x ) d x = x 3 arcsch ⁡ ( a x ) 3 − 1 6 a 3 arcoth ⁡ 1 a 2 x 2 + 1 + x 2 6 a 1 a 2 x 2 + 1 + C {\displaystyle \int x^{2}\,\operatorname {arcsch} (a\,x)dx={\frac {x^{3}\,\operatorname {arcsch} (a\,x)}{3}}\,-\,{\frac {1}{6\,a^{3}}}\,\operatorname {arcoth} {\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}\,+\,{\frac {x^{2}}{6\,a}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}\,+\,C} ∫ x m arcsch ⁡ ( a x ) d x = x m + 1 arcsch ⁡ ( a x ) m + 1 + 1 a ( m + 1 ) ∫ x m − 1 1 a 2 x 2 + 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arcsch} (a\,x)dx={\frac {x^{m+1}\operatorname {arcsch} (a\,x)}{m+1}}\,+\,{\frac {1}{a(m+1)}}\int {\frac {x^{m-1}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}}\,dx\quad (m\neq -1)}