Tavola degli integrali indefiniti di funzioni razionali

Questa pagina contiene una tavola di integrali indefiniti di funzioni razionali. C {\displaystyle C} denota una costante arbitraria di integrazione che ha senso specificare solo in relazione a una specificazione del valore dell'integrale in qualche punto.

Per altri integrali vedi Integrale § Tavole di integrali.
∫ ( a x + b ) n d x = ( a x + b ) n + 1 a ( n + 1 ) + C (per  n ≠ − 1 ) {\displaystyle \int (ax+b)^{n}\mathrm {d} x={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\text{(per }}n\neq -1{\text{)}}}
∫ x n − 1 ( a x n + b ) c d x = ( a x n + b ) c + 1 n a ( c + 1 ) + C (per  n ≠ 0 ) {\displaystyle \int x^{n-1}(ax^{n}+b)^{c}\;\mathrm {d} x={\frac {(ax^{n}+b)^{c+1}}{na(c+1)}}+C\qquad {\text{(per }}n\neq 0{\text{)}}}
∫ d x a x + b = 1 a ln ⁡ | a x + b | + C {\displaystyle \int {\frac {\mathrm {d} x}{ax+b}}={\frac {1}{a}}\ln \left|ax+b\right|+C}
∫ x d x a x + b = x a − b a 2 log ⁡ | a x + b | + C {\displaystyle \int {\frac {x\;\mathrm {d} x}{ax+b}}={\frac {x}{a}}-{\frac {b}{a^{2}}}\log \left|ax+b\right|+C}
∫ x d x ( a x + b ) 2 = b a 2 ( a x + b ) + 1 a 2 log ⁡ | a x + b | + C {\displaystyle \int {\frac {x\;\mathrm {d} x}{(ax+b)^{2}}}={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\log \left|ax+b\right|+C}
∫ x d x ( a x + b ) n = a ( 1 − n ) x − b a 2 ( n − 1 ) ( n − 2 ) ( a x + b ) n − 1 + C (per  n ∉ { 1 , 2 } ) {\displaystyle \int {\frac {x\;\mathrm {d} x}{(ax+b)^{n}}}={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}+C\qquad {\text{(per }}n\not \in \{1,2\}{\text{)}}}
∫ x 2 d x a x + b = 1 a 3 [ ( a x + b ) 2 2 − 2 b ( a x + b ) + b 2 log ⁡ | a x + b | ] + C {\displaystyle \int {\frac {x^{2}\;\mathrm {d} x}{ax+b}}={\frac {1}{a^{3}}}\left[{\frac {(ax+b)^{2}}{2}}-2b(ax+b)+b^{2}\log \left|ax+b\right|\right]+C}
∫ x 2 d x ( a x + b ) 2 = 1 a 3 ( a x + b − 2 b log ⁡ | a x + b | − b 2 a x + b ) + C {\displaystyle \int {\frac {x^{2}\;\mathrm {d} x}{(ax+b)^{2}}}={\frac {1}{a^{3}}}\left(ax+b-2b\log \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)+C}
∫ x 2 d x ( a x + b ) 3 = 1 a 3 [ log ⁡ | a x + b | + 2 b a x + b − b 2 2 ( a x + b ) 2 ] + C {\displaystyle \int {\frac {x^{2}\;\mathrm {d} x}{(ax+b)^{3}}}={\frac {1}{a^{3}}}\left[\log \left|ax+b\right|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right]+C}
∫ x 2 d x ( a x + b ) n = 1 a 3 [ − 1 ( n − 3 ) ( a x + b ) n − 3 + 2 b ( n − 2 ) ( a + b ) n − 2 − b 2 ( n − 1 ) ( a x + b ) n − 1 ] + C (per  n ∉ { 1 , 2 , 3 } ) {\displaystyle \int {\frac {x^{2}\;\mathrm {d} x}{(ax+b)^{n}}}={\frac {1}{a^{3}}}\left[-{\frac {1}{(n-3)(ax+b)^{n-3}}}+{\frac {2b}{(n-2)(a+b)^{n-2}}}-{\frac {b^{2}}{(n-1)(ax+b)^{n-1}}}\right]+C\qquad {\text{(per }}n\not \in \{1,2,3\}{\text{)}}}
∫ d x x ( a x + b ) = − 1 b log ⁡ | a x + b x | + C {\displaystyle \int {\frac {\mathrm {d} x}{x(ax+b)}}=-{\frac {1}{b}}\log \left|{\frac {ax+b}{x}}\right|+C}
∫ d x x 2 ( a x + b ) = − 1 b x + a b 2 log ⁡ | a x + b x | + C {\displaystyle \int {\frac {\mathrm {d} x}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\log \left|{\frac {ax+b}{x}}\right|+C}
∫ d x x 2 ( a x + b ) 2 = − a [ 1 b 2 ( a x + b ) + 1 a b 2 x − 2 b 3 log ⁡ | a x + b x | ] + C {\displaystyle \int {\frac {\mathrm {d} x}{x^{2}(ax+b)^{2}}}=-a\left[{\frac {1}{b^{2}(ax+b)}}+{\frac {1}{ab^{2}x}}-{\frac {2}{b^{3}}}\log \left|{\frac {ax+b}{x}}\right|\right]+C}
∫ d x x 2 + a 2 = 1 a arctan ⁡ x a + C {\displaystyle \int {\frac {\mathrm {d} x}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}+C}
∫ d x x 2 − a 2 = − 1 a s e t t a n h x a = 1 2 a log ⁡ a − x a + x + C (per  | x | < | a | ) {\displaystyle \int {\frac {\mathrm {d} x}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {settanh} {\frac {x}{a}}={\frac {1}{2a}}\log {\frac {a-x}{a+x}}+C\qquad {\text{(per }}|x|<|a|{\text{)}}}
∫ d x x 2 − a 2 = − 1 a s e t t c o t h x a = 1 2 a log ⁡ x − a x + a + C (per  | x | > | a | ) {\displaystyle \int {\frac {\mathrm {d} x}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {settcoth} {\frac {x}{a}}={\frac {1}{2a}}\log {\frac {x-a}{x+a}}+C\qquad {\text{(per }}|x|>|a|{\text{)}}}

Nelle formule che seguono si intende che sia a ≠ 0 {\displaystyle a\neq 0}

∫ d x a x 2 + b x + c = 2 4 a c − b 2 arctan ⁡ 2 a x + b 4 a c − b 2 + C (per  4 a c − b 2 > 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}+C\qquad {\text{(per }}4ac-b^{2}>0{\text{)}}}
∫ d x a x 2 + b x + c = − 2 2 a x + b + C (per  4 a c − b 2 = 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}=-{\frac {2}{2ax+b}}+C\qquad {\text{(per }}4ac-b^{2}=0{\text{)}}}
∫ d x a x 2 + b x + c = − 2 b 2 − 4 a c s e t t a n h 2 a x + b b 2 − 4 a c = 1 b 2 − 4 a c log ⁡ | 2 a x + b − b 2 − 4 a c 2 a x + b + b 2 − 4 a c | + C (per  4 a c − b 2 < 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}=-{\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {settanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}={\frac {1}{\sqrt {b^{2}-4ac}}}\log \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|+C\qquad {\mbox{(per }}4ac-b^{2}<0{\mbox{)}}}
∫ x d x a x 2 + b x + c = 1 2 a ln ⁡ | a x 2 + b x + c | − b 2 a ∫ d x a x 2 + b x + c {\displaystyle \int {\frac {x\;\mathrm {d} x}{ax^{2}+bx+c}}={\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 + C (per  4 a c − b 2 > 0 ) {\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}\mathrm {d} x={\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}}}}+C\qquad {\text{(per }}4ac-b^{2}>0{\text{)}}}
∫ 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 b 2 − 4 a c s e t t t a n h 2 a x + b b 2 − 4 a c + C (per  4 a c − b 2 < 0 ) {\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}\mathrm {d} x={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {setttanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+C\qquad {\text{(per }}4ac-b^{2}<0{\text{)}}}
∫ d x ( a x 2 + b x + c ) n = 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 ) ∫ d x ( a x 2 + b x + c ) n − 1 {\displaystyle \int {\frac {\mathrm {d} x}{(ax^{2}+bx+c)^{n}}}={\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 {\mathrm {d} x}{(ax^{2}+bx+c)^{n-1}}}}
∫ x d x ( a x 2 + b x + c ) n = − 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 ) ∫ d x ( a x 2 + b x + c ) n − 1 {\displaystyle \int {\frac {x\;\mathrm {d} x}{(ax^{2}+bx+c)^{n}}}=-{\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 {\mathrm {d} x}{(ax^{2}+bx+c)^{n-1}}}}
∫ d x x ( a x 2 + b x + c ) = 1 2 c log ⁡ | x 2 a x 2 + b x + c | − b 2 c ∫ d x a x 2 + b x + c {\displaystyle \int {\frac {\mathrm {d} x}{x(ax^{2}+bx+c)}}={\frac {1}{2c}}\log \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}}
∫ d x x 4 + 1 = 1 2 2 [ arctan ⁡ ( 2 x + 1 ) + arctan ⁡ ( 2 x − 1 ) ] + 1 4 2 [ log ⁡ | x 2 + 2 x + 1 | − log ⁡ | x 2 − 2 x + 1 | ] + C {\displaystyle \int {\frac {\mathrm {d} x}{x^{4}+1}}={\frac {1}{2{\sqrt {2}}}}\left[\arctan({\sqrt {2}}x+1)+\arctan({\sqrt {2}}x-1)\right]+{\frac {1}{4{\sqrt {2}}}}\left[\log |x^{2}+{\sqrt {2}}x+1|-\log |x^{2}-{\sqrt {2}}x+1|\right]+C}
∫ 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 ⋅ log ⁡ | x 2 − 2 x cos ⁡ ( 2 k − 1 ) π 2 n + 1 | } + C {\displaystyle \int {\frac {\mathrm {d} x}{x^{2^{n}}+1}}=\sum _{k=1}^{2^{n-1}}\left\{{\frac {1}{2^{n-1}}}\sin {\frac {(2k-1)\pi }{2^{n}}}\cdot \arctan \left[\left(x-\cos {\frac {(2k-1)\pi }{2^{n}}}\right)\csc {\frac {(2k-1)\pi }{2^{n}}}\right]-{\frac {1}{2^{n}}}\cos {\frac {(2k-1)\pi }{2^{n}}}\cdot \log \left|x^{2}-2x\cos {\frac {(2k-1)\pi }{2^{n}}}+1\right|\right\}+C}
∫ d x x n + 1 = x 2 F 1 ( 1 , 1 n ; 1 + 1 n ; − x n ) + C {\displaystyle \int {\frac {\mathrm {d} x}{x^{n}+1}}={x}_{2}F_{1}\left({1,{\frac {1}{n}};1+{\frac {1}{n}}};-x^{n}\right)+C} [1]

dove p F q ( a 1 , … , a p ; b 1 , … , b q ; z ) {\displaystyle {}_{p}F_{q}\left({a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q}};z\right)} indica la serie ipergeometrica.

Di ogni funzione razionale si riesce a trovare l'integrale indefinito decomponendola in una somma di funzioni della forma

e x + f ( a x 2 + b x + c ) n {\displaystyle {\frac {ex+f}{\left(ax^{2}+bx+c\right)^{n}}}}

e applicando ai diversi addendi qualcuna delle formule precedenti.

Note

  1. ^ (EN) Integrate[Divide[1,1 Power[x,n]],x] - Wolfram|Alpha, su www.wolframalpha.com. URL consultato il 19 aprile 2023.

Bibliografia

  • Murray R. Spiegel, Manuale di matematica, Etas Libri, 1974, pp. 60-74.
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