無理関数の原始関数の一覧
∫ x r 2 n + 1 d x = r 2 n + 3 2 n + 3 {\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}}
∫ x 2 r d x = x r 3 4 − a 2 x r 8 − a 4 8 ln  ( x + r ) {\displaystyle \int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left(x+r\right)}
∫ x 2 r 3 d x = x r 5 6 − a 2 x r 3 24 − a 4 x r 16 − a 6 16 ln  ( x + r ) {\displaystyle \int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left(x+r\right)}
∫ x 3 r d x = r 5 5 − a 2 r 3 3 {\displaystyle \int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}
∫ x 3 r 3 d x = r 7 7 − a 2 r 5 5 {\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}}
∫ r 3 d x x = r 3 3 + a 2 r − a 3 ln  | a + r x | {\displaystyle \int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}
∫ r 5 d x x = r 5 5 + a 2 r 3 3 + a 4 r − a 5 ln  | a + r x | {\displaystyle \int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}
∫ r 7 d x x = r 7 7 + a 2 r 5 5 + a 4 r 3 3 + a 6 r − a 7 ln  | a + r x | {\displaystyle \int {\frac {r^{7}\;dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}
∫ d x r = arsinh  x a = ln  ( x + r a ) {\displaystyle \int {\frac {dx}{r}}=\operatorname {arsinh} {\frac {x}{a}}=\ln \left({\frac {x+r}{a}}\right)}