無理関数の原始関数の一覧
∫ x 3 r 2 n + 1 d x = r 2 n + 5 2 n + 5 − a 3 r 2 n + 3 2 n + 3 {\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}}
∫ x 4 r d x = x 3 r 3 6 − a 2 x r 3 8 + a 4 x r 16 + a 6 16 ln  ( x + r ) {\displaystyle \int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left(x+r\right)}
∫ x 4 r 3 d x = x 3 r 5 8 − a 2 x r 5 16 + a 4 x r 3 64 + 3 a 6 x r 128 + 3 a 8 128 ln  ( x + r ) {\displaystyle \int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left(x+r\right)}
∫ d x r 3 = x a 2 r {\displaystyle \int {\frac {dx}{r^{3}}}={\frac {x}{a^{2}r}}}
∫ x d x r = r {\displaystyle \int {\frac {x\,dx}{r}}=r}
∫ x d x r 3 = − 1 r {\displaystyle \int {\frac {x\,dx}{r^{3}}}=-{\frac {1}{r}}}
∫ x 2 d x r = x 2 r − a 2 2 arsinh  x a = x 2 r − a 2 2 ln  ( x + r a ) {\displaystyle \int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\operatorname {arsinh} {\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left({\frac {x+r}{a}}\right)}
∫ d x x r = − 1 a arsinh  a x = − 1 a ln  | a + r x | {\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\operatorname {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}
∫ d x R = − 1 − a arcsin  2 a x + b b 2 − 4 a c (for a < 0 , 4 a c − b 2 < 0 , | 2 a x + b | < b 2 − 4 a c ) {\displaystyle \int {\frac {dx}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(for }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{, }}\left|2ax+b\right|<{\sqrt {b^{2}-4ac}}{\mbox{)}}}
∫ d x R 3 = 4 a x + 2 b ( 4 a c − b 2 ) R {\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}