有理関数の原始関数の一覧
本項は、有理関数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。
∫ ( a x + b ) n d x = ( a x + b ) n + 1 a ( n + 1 ) + C (for n ≠ − 1 ) {\displaystyle \int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
∫ c a x + b d x = c a ln  | a x + b | + C {\displaystyle \int {\frac {c}{ax+b}}dx={\frac {c}{a}}\ln \left|ax+b\right|+C}
∫ x ( a x + b ) n d x = a ( n + 1 ) x − b a 2 ( n + 1 ) ( n + 2 ) ( a x + b ) n + 1 + C (for n ∉ { − 1 , − 2 } ) {\displaystyle \int x(ax+b)^{n}dx={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}+C\qquad {\mbox{(for }}n\not \in \{-1,-2\}{\mbox{)}}}
∫ x a x + b d x = x a − b a 2 ln  | a x + b | + C {\displaystyle \int {\frac {x}{ax+b}}dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|+C}
∫ x ( a x + b ) 2 d x = b a 2 ( a x + b ) + 1 a 2 ln  | a x + b | + C {\displaystyle \int {\frac {x}{(ax+b)^{2}}}dx={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|+C}
∫ x ( a x + b ) n d x = a ( 1 − n ) x − b a 2 ( n − 1 ) ( n − 2 ) ( a x + b ) n − 1 + C (for n ∉ { 1 , 2 } ) {\displaystyle \int {\frac {x}{(ax+b)^{n}}}dx={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}+C\qquad {\mbox{(for }}n\not \in \{1,2\}{\mbox{)}}}
∫ f ′ ( x ) f ( x ) d x = ln  | f ( x ) | + C {\displaystyle \int {\frac {f'(x)}{f(x)}}dx=\ln \left|f(x)\right|+C}
∫ x 2 a x + b d x = b 2 ln  ( | a x + b | ) a 3 + a x 2 − 2 b x 2 a 2 + C {\displaystyle \int {\frac {x^{2}}{ax+b}}dx={\frac {b^{2}\ln(\left|ax+b\right|)}{a^{3}}}+{\frac {ax^{2}-2bx}{2a^{2}}}+C}