逆三角関数の原始関数の一覧

出典: フリー百科事典『ウィキペディア(Wikipedia)』

本項は逆三角関数を含む式の原始関数の一覧である。さらに完全な原始関数の一覧は、原始関数の一覧を参照のこと。

以下の全ての記述において、a は 0 でない実数とする。また、C は積分定数とする。

${\displaystyle \int \arcsin x,円dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}${\displaystyle \int \arcsin x,円dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}

${\displaystyle \int \arcsin ax,円dx=x\arcsin ax+{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}${\displaystyle \int \arcsin ax,円dx=x\arcsin ax+{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}

${\displaystyle \int x\arcsin ax,円dx={\frac {x^{2}\arcsin ax}{2}}-{\frac {\arcsin ax}{4a^{2}}}+{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4,円a}}+C}${\displaystyle \int x\arcsin ax,円dx={\frac {x^{2}\arcsin ax}{2}}-{\frac {\arcsin ax}{4a^{2}}}+{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4,円a}}+C}

${\displaystyle \int x^{2}\arcsin ax,円dx={\frac {x^{3}\arcsin ax}{3}}+{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}}}}{9,円a^{3}}}+C}${\displaystyle \int x^{2}\arcsin ax,円dx={\frac {x^{3}\arcsin ax}{3}}+{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}}}}{9,円a^{3}}}+C}

${\displaystyle \int x^{m}\arcsin ax,円dx={\frac {x^{m+1}\arcsin ax}{m+1}},円-,円{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}x^{2}}}},円dx\quad (m\neq -1)}${\displaystyle \int x^{m}\arcsin ax,円dx={\frac {x^{m+1}\arcsin ax}{m+1}},円-,円{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}x^{2}}}},円dx\quad (m\neq -1)}

${\displaystyle \int (\arcsin ax)^{2},円dx=-2,円x+x,円(\arcsin ax)^{2}+{\frac {2{\sqrt {1-a^{2}x^{2}}}\arcsin ax}{a}}+C}${\displaystyle \int (\arcsin ax)^{2},円dx=-2,円x+x,円(\arcsin ax)^{2}+{\frac {2{\sqrt {1-a^{2}x^{2}}}\arcsin ax}{a}}+C}

${\displaystyle \int (\arcsin ax)^{n},円dx=x,円(\arcsin ax)^{n},円+,円{\frac {n{\sqrt {1-a^{2}x^{2}}},円(\arcsin ax)^{n-1}}{a}},円-,円n,円(n-1)\int (\arcsin ax)^{n-2},円dx}${\displaystyle \int (\arcsin ax)^{n},円dx=x,円(\arcsin ax)^{n},円+,円{\frac {n{\sqrt {1-a^{2}x^{2}}},円(\arcsin ax)^{n-1}}{a}},円-,円n,円(n-1)\int (\arcsin ax)^{n-2},円dx}

${\displaystyle \int (\arcsin ax)^{n},円dx={\frac {x,円(\arcsin ax)^{n+2}}{(n+1),円(n+2)}},円+,円{\frac {{\sqrt {1-a^{2}x^{2}}},円(\arcsin ax)^{n+1}}{a,円(n+1)}},円-,円{\frac {1}{(n+1),円(n+2)}}\int (\arcsin ax)^{n+2},円dx\quad (n\neq -1,-2)}${\displaystyle \int (\arcsin ax)^{n},円dx={\frac {x,円(\arcsin ax)^{n+2}}{(n+1),円(n+2)}},円+,円{\frac {{\sqrt {1-a^{2}x^{2}}},円(\arcsin ax)^{n+1}}{a,円(n+1)}},円-,円{\frac {1}{(n+1),円(n+2)}}\int (\arcsin ax)^{n+2},円dx\quad (n\neq -1,-2)}

${\displaystyle \int \arccos x,円dx=x\arccos x-{\sqrt {1-x^{2}}}+C}${\displaystyle \int \arccos x,円dx=x\arccos x-{\sqrt {1-x^{2}}}+C}

${\displaystyle \int \arccos ax,円dx=x\arccos ax-{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}${\displaystyle \int \arccos ax,円dx=x\arccos ax-{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}

${\displaystyle \int x\arccos ax,円dx={\frac {x^{2}\arccos ax}{2}}-{\frac {\arccos ax}{4,円a^{2}}}-{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4,円a}}+C}${\displaystyle \int x\arccos ax,円dx={\frac {x^{2}\arccos ax}{2}}-{\frac {\arccos ax}{4,円a^{2}}}-{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4,円a}}+C}

${\displaystyle \int x^{2}\arccos ax,円dx={\frac {x^{3}\arccos ax}{3}}-{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}}}}{9,円a^{3}}}+C}${\displaystyle \int x^{2}\arccos ax,円dx={\frac {x^{3}\arccos ax}{3}}-{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}}}}{9,円a^{3}}}+C}

${\displaystyle \int x^{m}\arccos ax,円dx={\frac {x^{m+1}\arccos ax}{m+1}},円+,円{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}x^{2}}}},円dx\quad (m\neq -1)}${\displaystyle \int x^{m}\arccos ax,円dx={\frac {x^{m+1}\arccos ax}{m+1}},円+,円{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}x^{2}}}},円dx\quad (m\neq -1)}

${\displaystyle \int (\arccos ax)^{2},円dx=-2,円x+x,円(\arccos ax)^{2}-{\frac {2{\sqrt {1-a^{2}x^{2}}}\arccos ax}{a}}+C}${\displaystyle \int (\arccos ax)^{2},円dx=-2,円x+x,円(\arccos ax)^{2}-{\frac {2{\sqrt {1-a^{2}x^{2}}}\arccos ax}{a}}+C}

${\displaystyle \int (\arccos ax)^{n},円dx=x,円(\arccos ax)^{n},円-,円{\frac {n{\sqrt {1-a^{2}x^{2}}},円(\arccos ax)^{n-1}}{a}},円-,円n,円(n-1)\int (\arccos ax)^{n-2},円dx}${\displaystyle \int (\arccos ax)^{n},円dx=x,円(\arccos ax)^{n},円-,円{\frac {n{\sqrt {1-a^{2}x^{2}}},円(\arccos ax)^{n-1}}{a}},円-,円n,円(n-1)\int (\arccos ax)^{n-2},円dx}

${\displaystyle \int (\arccos ax)^{n},円dx={\frac {x,円(\arccos ax)^{n+2}}{(n+1),円(n+2)}},円-,円{\frac {{\sqrt {1-a^{2}x^{2}}},円(\arccos ax)^{n+1}}{a,円(n+1)}},円-,円{\frac {1}{(n+1),円(n+2)}}\int (\arccos ax)^{n+2},円dx\quad (n\neq -1,-2)}${\displaystyle \int (\arccos ax)^{n},円dx={\frac {x,円(\arccos ax)^{n+2}}{(n+1),円(n+2)}},円-,円{\frac {{\sqrt {1-a^{2}x^{2}}},円(\arccos ax)^{n+1}}{a,円(n+1)}},円-,円{\frac {1}{(n+1),円(n+2)}}\int (\arccos ax)^{n+2},円dx\quad (n\neq -1,-2)}

${\displaystyle \int \arctan x,円dx=x\arctan x-{\frac {\ln(x^{2}+1)}{2}}+C}${\displaystyle \int \arctan x,円dx=x\arctan x-{\frac {\ln(x^{2}+1)}{2}}+C}

${\displaystyle \int \arctan ax,円dx=x\arctan ax-{\frac {\ln(a^{2}x^{2}+1)}{2,円a}}+C}${\displaystyle \int \arctan ax,円dx=x\arctan ax-{\frac {\ln(a^{2}x^{2}+1)}{2,円a}}+C}

${\displaystyle \int x\arctan ax,円dx={\frac {x^{2}\arctan ax}{2}}+{\frac {\arctan ax}{2,円a^{2}}}-{\frac {x}{2,円a}}+C}${\displaystyle \int x\arctan ax,円dx={\frac {x^{2}\arctan ax}{2}}+{\frac {\arctan ax}{2,円a^{2}}}-{\frac {x}{2,円a}}+C}

${\displaystyle \int x^{2}\arctan ax,円dx={\frac {x^{3}\arctan ax}{3}}+{\frac {\ln(a^{2}x^{2}+1)}{6,円a^{3}}}-{\frac {x^{2}}{6,円a}}+C}${\displaystyle \int x^{2}\arctan ax,円dx={\frac {x^{3}\arctan ax}{3}}+{\frac {\ln(a^{2}x^{2}+1)}{6,円a^{3}}}-{\frac {x^{2}}{6,円a}}+C}

${\displaystyle \int x^{m}\arctan ax,円dx={\frac {x^{m+1}\arctan ax}{m+1}}-{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}x^{2}+1}},円dx\quad (m\neq -1)}${\displaystyle \int x^{m}\arctan ax,円dx={\frac {x^{m+1}\arctan ax}{m+1}}-{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}x^{2}+1}},円dx\quad (m\neq -1)}

${\displaystyle \int \operatorname {arccot} x,円dx=x\operatorname {arccot} x+{\frac {\ln \left(x^{2}+1\right)}{2}}+C}${\displaystyle \int \operatorname {arccot} x,円dx=x\operatorname {arccot} x+{\frac {\ln \left(x^{2}+1\right)}{2}}+C}

${\displaystyle \int \operatorname {arccot} ax,円dx=x\operatorname {arccot} ax+{\frac {\ln \left(a^{2}x^{2}+1\right)}{2,円a}}+C}${\displaystyle \int \operatorname {arccot} ax,円dx=x\operatorname {arccot} ax+{\frac {\ln \left(a^{2}x^{2}+1\right)}{2,円a}}+C}

${\displaystyle \int x\operatorname {arccot} ax,円dx={\frac {x^{2}\operatorname {arccot} ax}{2}}+{\frac {\operatorname {arccot} ax}{2,円a^{2}}}+{\frac {x}{2,円a}}+C}${\displaystyle \int x\operatorname {arccot} ax,円dx={\frac {x^{2}\operatorname {arccot} ax}{2}}+{\frac {\operatorname {arccot} ax}{2,円a^{2}}}+{\frac {x}{2,円a}}+C}

${\displaystyle \int x^{2}\operatorname {arccot} ax,円dx={\frac {x^{3}\operatorname {arccot} ax}{3}}-{\frac {\ln \left(a^{2}x^{2}+1\right)}{6,円a^{3}}}+{\frac {x^{2}}{6,円a}}+C}${\displaystyle \int x^{2}\operatorname {arccot} ax,円dx={\frac {x^{3}\operatorname {arccot} ax}{3}}-{\frac {\ln \left(a^{2}x^{2}+1\right)}{6,円a^{3}}}+{\frac {x^{2}}{6,円a}}+C}

${\displaystyle \int x^{m}\operatorname {arccot} ax,円dx={\frac {x^{m+1}\operatorname {arccot} ax}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}x^{2}+1}},円dx\quad (m\neq -1)}${\displaystyle \int x^{m}\operatorname {arccot} ax,円dx={\frac {x^{m+1}\operatorname {arccot} ax}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}x^{2}+1}},円dx\quad (m\neq -1)}

${\displaystyle \int \operatorname {arcsec} x,円dx=x\operatorname {arcsec} x-\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{x^{2}}}}}+C}${\displaystyle \int \operatorname {arcsec} x,円dx=x\operatorname {arcsec} x-\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{x^{2}}}}}+C}

${\displaystyle \int \operatorname {arcsec} ax,円dx=x\operatorname {arcsec} ax-{\frac {1}{a}},円\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}${\displaystyle \int \operatorname {arcsec} ax,円dx=x\operatorname {arcsec} ax-{\frac {1}{a}},円\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}

${\displaystyle \int x\operatorname {arcsec} ax,円dx={\frac {x^{2}\operatorname {arcsec} ax}{2}}-{\frac {x}{2,円a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}${\displaystyle \int x\operatorname {arcsec} ax,円dx={\frac {x^{2}\operatorname {arcsec} ax}{2}}-{\frac {x}{2,円a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}

${\displaystyle \int x^{2}\operatorname {arcsec} ax,円dx={\frac {x^{3}\operatorname {arcsec} ax}{3}},円-,円{\frac {1}{6,円a^{3}}},円\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}},円-,円{\frac {x^{2}}{6,円a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}},円+,円C}${\displaystyle \int x^{2}\operatorname {arcsec} ax,円dx={\frac {x^{3}\operatorname {arcsec} ax}{3}},円-,円{\frac {1}{6,円a^{3}}},円\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}},円-,円{\frac {x^{2}}{6,円a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}},円+,円C}

${\displaystyle \int x^{m}\operatorname {arcsec} ax,円dx={\frac {x^{m+1}\operatorname {arcsec} ax}{m+1}},円-,円{\frac {1}{a,円(m+1)}}\int {\frac {x^{m-1}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}},円dx\quad (m\neq -1)}${\displaystyle \int x^{m}\operatorname {arcsec} ax,円dx={\frac {x^{m+1}\operatorname {arcsec} ax}{m+1}},円-,円{\frac {1}{a,円(m+1)}}\int {\frac {x^{m-1}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}},円dx\quad (m\neq -1)}

${\displaystyle \int \operatorname {arccsc} x,円dx=x\operatorname {arccsc} x,円+,円\ln \left|x+{\sqrt {x^{2}-1}}\right|,円+,円C=x\operatorname {arccsc} x,円+,円\operatorname {arccosh} (x),円+,円C}${\displaystyle \int \operatorname {arccsc} x,円dx=x\operatorname {arccsc} x,円+,円\ln \left|x+{\sqrt {x^{2}-1}}\right|,円+,円C=x\operatorname {arccsc} x,円+,円\operatorname {arccosh} (x),円+,円C}

${\displaystyle \int \operatorname {arccsc} ax,円dx=x\operatorname {arccsc} ax+{\frac {1}{a}},円\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}${\displaystyle \int \operatorname {arccsc} ax,円dx=x\operatorname {arccsc} ax+{\frac {1}{a}},円\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}

${\displaystyle \int x\operatorname {arccsc} ax,円dx={\frac {x^{2}\operatorname {arccsc} ax}{2}}+{\frac {x}{2,円a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}${\displaystyle \int x\operatorname {arccsc} ax,円dx={\frac {x^{2}\operatorname {arccsc} ax}{2}}+{\frac {x}{2,円a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}

${\displaystyle \int x^{2}\operatorname {arccsc} ax,円dx={\frac {x^{3}\operatorname {arccsc} ax}{3}},円+,円{\frac {1}{6,円a^{3}}},円\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}},円+,円{\frac {x^{2}}{6,円a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}},円+,円C}${\displaystyle \int x^{2}\operatorname {arccsc} ax,円dx={\frac {x^{3}\operatorname {arccsc} ax}{3}},円+,円{\frac {1}{6,円a^{3}}},円\operatorname {arctanh} ,円{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}},円+,円{\frac {x^{2}}{6,円a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}},円+,円C}

${\displaystyle \int x^{m}\operatorname {arccsc} ax,円dx={\frac {x^{m+1}\operatorname {arccsc} ax}{m+1}},円+,円{\frac {1}{a,円(m+1)}}\int {\frac {x^{m-1}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}},円dx\quad (m\neq -1)}${\displaystyle \int x^{m}\operatorname {arccsc} ax,円dx={\frac {x^{m+1}\operatorname {arccsc} ax}{m+1}},円+,円{\frac {1}{a,円(m+1)}}\int {\frac {x^{m-1}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}},円dx\quad (m\neq -1)}

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