Lijst van integralen van rationale functies

Hieronder staat een lijst van integralen van rationale functies. Integralen zijn het onderwerp van studie van de integraalrekening. Een rationale functie is een breuk waarvan zowel de teller als de noemer een polynoom is of gelijk is aan 1.

∫ ( a x + b ) n   d x = ( a x + b ) n + 1 a ( n + 1 ) voor  n ≠ − 1 {\displaystyle \int (ax+b)^{n}\ \mathrm {d} x={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\mbox{voor }}n\neq -1}


∫ 1 a x + b   d x = 1 a ln ⁡ | a x + b | {\displaystyle \int {\frac {1}{ax+b}}\ \mathrm {d} x={\frac {1}{a}}\ln |ax+b|}


∫ x ( a x + b ) n   d x = a ( n + 1 ) x − b a 2 ( n + 1 ) ( n + 2 ) ( a x + b ) n + 1 voor  n ∉ { − 1 , − 2 } {\displaystyle \int x(ax+b)^{n}\ \mathrm {d} x={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad {\mbox{voor }}n\not \in \{-1,-2\}}


∫ x a x + b   d x = x a − b a 2 ln ⁡ | a x + b | {\displaystyle \int {\frac {x}{ax+b}}\ \mathrm {d} x={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln |ax+b|}


∫ x ( a x + b ) 2   d x = b a 2 ( a x + b ) + 1 a 2 ln ⁡ | a x + b | {\displaystyle \int {\frac {x}{(ax+b)^{2}}}\ \mathrm {d} x={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln |ax+b|}


∫ x ( a x + b ) n   d x = a ( 1 − n ) x − b a 2 ( n − 1 ) ( n − 2 ) ( a x + b ) n − 1 voor  n ∉ { 1 , 2 } {\displaystyle \int {\frac {x}{(ax+b)^{n}}}\ \mathrm {d} x={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}\qquad {\mbox{voor }}n\not \in \{1,2\}}


∫ x 2 a x + b   d x = b 2 ln ⁡ | a x + b | a 3 + a x 2 − 2 b x 2 a 2 {\displaystyle \int {\frac {x^{2}}{ax+b}}\ \mathrm {d} x={\frac {b^{2}\ln |ax+b|}{a^{3}}}+{\frac {ax^{2}-2bx}{2a^{2}}}}


∫ x 2 ( a x + b ) 2   d x = 1 a 3 ( a x − 2 b ln ⁡ | a x + b | − b 2 a x + b ) {\displaystyle \int {\frac {x^{2}}{(ax+b)^{2}}}\ \mathrm {d} x={\frac {1}{a^{3}}}\left(ax-2b\ln |ax+b|-{\frac {b^{2}}{ax+b}}\right)}


∫ x 2 ( a x + b ) 3   d x = 1 a 3 ( ln ⁡ | a x + b | + 2 b a x + b − b 2 2 ( a x + b ) 2 ) {\displaystyle \int {\frac {x^{2}}{(ax+b)^{3}}}\ \mathrm {d} x={\frac {1}{a^{3}}}\left(\ln |ax+b|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right)}


∫ x 2 ( a x + b ) n   d x = 1 a 3 ( − ( a x + b ) 3 − n ( n − 3 ) + 2 b ( a + b ) 2 − n ( n − 2 ) − b 2 ( a x + b ) 1 − n ( n − 1 ) ) voor  n ∉ { 1 , 2 , 3 } {\displaystyle \int {\frac {x^{2}}{(ax+b)^{n}}}\ \mathrm {d} x={\frac {1}{a^{3}}}\left(-{\frac {(ax+b)^{3-n}}{(n-3)}}+{\frac {2b(a+b)^{2-n}}{(n-2)}}-{\frac {b^{2}(ax+b)^{1-n}}{(n-1)}}\right)\qquad {\mbox{voor }}n\not \in \{1,2,3\}}


∫ 1 x ( a x + b )   d x = − 1 b ln ⁡ | a x + b x | {\displaystyle \int {\frac {1}{x(ax+b)}}\ \mathrm {d} x=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|}


∫ 1 x 2 ( a x + b )   d x = − 1 b x + a b 2 ln ⁡ | a x + b x | {\displaystyle \int {\frac {1}{x^{2}(ax+b)}}\ \mathrm {d} x=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|}


∫ 1 x 2 ( a x + b ) 2   d x = − a ( 1 b 2 ( a x + b ) + 1 a b 2 x − 2 b 3 ln ⁡ | a x + b x | ) {\displaystyle \int {\frac {1}{x^{2}(ax+b)^{2}}}\ \mathrm {d} x=-a\left({\frac {1}{b^{2}(ax+b)}}+{\frac {1}{ab^{2}x}}-{\frac {2}{b^{3}}}\ln \left|{\frac {ax+b}{x}}\right|\right)}


∫ 1 x 2 + a 2   d x = 1 a arctan ⁡ x a {\displaystyle \int {\frac {1}{x^{2}+a^{2}}}\ \mathrm {d} x={\frac {1}{a}}\arctan {\frac {x}{a}}}


∫ 1 x 2 − a 2   d x = { − 1 a   a r c t a n h x a = 1 2 a ln ⁡ a − x a + x voor  | x | < | a | − 1 a   a r c c o t h x a = 1 2 a ln ⁡ x − a x + a voor  | x | > | a | {\displaystyle \int {\frac {1}{x^{2}-a^{2}}}\ \mathrm {d} x={\begin{cases}-{\frac {1}{a}}\ \mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}&{\mbox{voor }}|x|<|a|\\-{\frac {1}{a}}\ \mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}&{\mbox{voor }}|x|>|a|\end{cases}}}


∫   d x x 2 n + 1 = ∑ k = 1 2 n − 1 { 1 2 n − 1 [ sin ⁡ ( ( 2 k − 1 ) π 2 n ) arctan ⁡ [ ( x − cos ⁡ ( ( 2 k − 1 ) π 2 n ) ) csc ⁡ ( ( 2 k − 1 ) π 2 n ) ] ] − 1 2 n [ cos ⁡ ( ( 2 k − 1 ) π 2 n ) ln ⁡ | x 2 − 2 x cos ⁡ ( ( 2 k − 1 ) π 2 n ) + 1 | ] } {\displaystyle \int {\frac {\ \mathrm {d} x}{x^{2^{n}}+1}}=\sum _{k=1}^{2^{n-1}}\left\{{\frac {1}{2^{n-1}}}\left[\sin({\frac {(2k-1)\pi }{2^{n}}})\arctan[\left(x-\cos({\frac {(2k-1)\pi }{2^{n}}})\right)\csc({\frac {(2k-1)\pi }{2^{n}}})]\right]-{\frac {1}{2^{n}}}\left[\cos({\frac {(2k-1)\pi }{2^{n}}})\ln \left|x^{2}-2x\cos({\frac {(2k-1)\pi }{2^{n}}})+1\right|\right]\right\}}


voor a ≠ 0


∫ 1 a x 2 + b x + c   d x = { 2 4 a c − b 2 arctan ⁡ 2 a x + b 4 a c − b 2 voor  4 a c − b 2 > 0 − 2 b 2 − 4 a c   a r c t a n h 2 a x + b b 2 − 4 a c = 1 b 2 − 4 a c ln ⁡ | 2 a x + b − b 2 − 4 a c 2 a x + b + b 2 − 4 a c | voor  4 a c − b 2 < 0 − 2 2 a x + b voor  4 a c − b 2 = 0 {\displaystyle \int {\frac {1}{ax^{2}+bx+c}}\ \mathrm {d} x={\begin{cases}{\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}&{\mbox{voor }}4ac-b^{2}>0\\-{\frac {2}{\sqrt {b^{2}-4ac}}}\ \mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}={\frac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|&{\mbox{voor }}4ac-b^{2}<0\\-{\frac {2}{2ax+b}}&{\mbox{voor }}4ac-b^{2}=0\end{cases}}}


∫ x a x 2 + b x + c   d x {\displaystyle \int {\frac {x}{ax^{2}+bx+c}}\ \mathrm {d} x} || = 1 2 a ln ⁡ | a x 2 + b x + c | − b 2 a ∫   d x a x 2 + b x + c {\displaystyle ={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {\ \mathrm {d} x}{ax^{2}+bx+c}}}


∫ m x + n a x 2 + b x + c   d x = { m 2 a ln ⁡ | a x 2 + b x + c | + 2 a n − b m a 4 a c − b 2 arctan ⁡ 2 a x + b 4 a c − b 2 voor  4 a c − b 2 > 0 m 2 a ln ⁡ | a x 2 + b x + c ‖ − 2 a n − b m a b 2 − 4 a c   a r c t a n h 2 a x + b b 2 − 4 a c voor  4 a c − b 2 < 0 m 2 a ln ⁡ | a x 2 + b x + c | − 2 a n − b m a ( 2 a x + b ) voor  4 a c − b 2 = 0 {\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}\ \mathrm {d} x={\begin{cases}{\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {4ac-b^{2}}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}&{\mbox{voor }}4ac-b^{2}>0\\{\frac {m}{2a}}\ln |ax^{2}+bx+c\|-{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\ \mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}&{\mbox{voor }}4ac-b^{2}<0\\{\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}&{\mbox{voor }}4ac-b^{2}=0\end{cases}}}


∫ 1 ( a x 2 + b x + c ) n   d x = 2 a x + b ( n − 1 ) ( 4 a c − b 2 ) ( a x 2 + b x + c ) n − 1 + ( 2 n − 3 ) 2 a ( n − 1 ) ( 4 a c − b 2 ) ∫ 1 ( a x 2 + b x + c ) n − 1   d x {\displaystyle \int {\frac {1}{(ax^{2}+bx+c)^{n}}}\ \mathrm {d} x={\frac {2ax+b}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}+{\frac {(2n-3)2a}{(n-1)(4ac-b^{2})}}\int {\frac {1}{(ax^{2}+bx+c)^{n-1}}}\ \mathrm {d} x}


∫ x ( a x 2 + b x + c ) n   d x = − b x + 2 c ( n − 1 ) ( 4 a c − b 2 ) ( a x 2 + b x + c ) n − 1 − b ( 2 n − 3 ) ( n − 1 ) ( 4 a c − b 2 ) ∫ 1 ( a x 2 + b x + c ) n − 1   d x {\displaystyle \int {\frac {x}{(ax^{2}+bx+c)^{n}}}\ \mathrm {d} x=-{\frac {bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}-{\frac {b(2n-3)}{(n-1)(4ac-b^{2})}}\int {\frac {1}{(ax^{2}+bx+c)^{n-1}}}\ \mathrm {d} x}


∫ 1 x ( a x 2 + b x + c )   d x = 1 2 c ln ⁡ | x 2 a x 2 + b x + c | − b 2 c ∫ 1 a x 2 + b x + c   d x {\displaystyle \int {\frac {1}{x(ax^{2}+bx+c)}}\ \mathrm {d} x={\frac {1}{2c}}\ln \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {1}{ax^{2}+bx+c}}\ \mathrm {d} x}


met een wortel in de noemer


∫ a x n b c x m   d x = a x n + 1 b ( − m 2 + n + 1 ) c x m {\displaystyle \int {\frac {ax^{n}}{b{\sqrt {cx^{m}}}}}\ \mathrm {d} x={\frac {ax^{n+1}}{b\left(-{\frac {m}{2}}+n+1\right){\sqrt {cx^{m}}}}}}


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Lijst van integralen