Tavola degli integrali indefiniti di funzioni irrazionali

Questa pagina contiene una tavola di integrali indefiniti di funzioni irrazionali. Per altri integrali vedi Tavole di Integrali.

∫ a 2 − x 2 d x = 1 2 ( x a 2 − x 2 + a 2 arcsin ⁡ x a ) ( | x | ≤ | a | ) {\displaystyle \int {\sqrt {a^{2}-x^{2}}}\;\mathrm {d} x={\frac {1}{2}}\left(x{\sqrt {a^{2}-x^{2}}}+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫ x a 2 − x 2 d x = − 1 3 ( a 2 − x 2 ) 3 ( | x | ≤ | a | ) {\displaystyle \int x{\sqrt {a^{2}-x^{2}}}\;\mathrm {d} x=-{\frac {1}{3}}{\sqrt {(a^{2}-x^{2})^{3}}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫ a 2 − x 2 d x x = a 2 − x 2 − a log ⁡ | a + a 2 + x 2 x | ( | x | ≤ | a | ) {\displaystyle \int {\frac {{\sqrt {a^{2}-x^{2}}}\;\mathrm {d} x}{x}}={\sqrt {a^{2}-x^{2}}}-a\log \left|{\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫ d x a 2 − x 2 = arcsin ⁡ x a ( | x | ≤ | a | ) {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫ x 2 d x a 2 − x 2 = − x 2 a 2 − x 2 + a 2 2 arcsin ⁡ x a ( | x | ≤ | a | ) {\displaystyle \int {\frac {x^{2}\;\mathrm {d} x}{\sqrt {a^{2}-x^{2}}}}=-{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫ x 2 + a 2 d x = 1 2 ( x x 2 + a 2 + a 2 log ⁡ ( x + x 2 + a 2 ) ) = 1 2 ( x x 2 + a 2 + a 2 a r s i n h x a ) {\displaystyle \int {\sqrt {x^{2}+a^{2}}}\;\mathrm {d} x={\frac {1}{2}}\left(x{\sqrt {x^{2}+a^{2}}}+a^{2}\,\log \left(x+{\sqrt {x^{2}+a^{2}}}\right)\right)={\frac {1}{2}}\left(x{\sqrt {x^{2}+a^{2}}}+a^{2}\,\mathrm {arsinh} {\frac {x}{a}}\right)}
∫ x x 2 + a 2 d x = 1 3 ( x 2 + a 2 ) 3 {\displaystyle \int x{\sqrt {x^{2}+a^{2}}}\;\mathrm {d} x={\frac {1}{3}}{\sqrt {(x^{2}+a^{2})^{3}}}}
∫ x 2 + a 2 d x x = x 2 + a 2 − a log ⁡ | a + x 2 + a 2 x | {\displaystyle \int {\frac {{\sqrt {x^{2}+a^{2}}}\;\mathrm {d} x}{x}}={\sqrt {x^{2}+a^{2}}}-a\log \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|}
∫ d x x 2 + a 2 = a r s i n h x a = log ⁡ | x + x 2 + a 2 | {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {x^{2}+a^{2}}}}=\mathrm {arsinh} {\frac {x}{a}}=\log \left|x+{\sqrt {x^{2}+a^{2}}}\right|}
∫ x d x x 2 + a 2 = x 2 + a 2 {\displaystyle \int {\frac {x\,\mathrm {d} x}{\sqrt {x^{2}+a^{2}}}}={\sqrt {x^{2}+a^{2}}}}
∫ x 2 d x x 2 + a 2 = x 2 x 2 + a 2 − a 2 2 a r s i n h x a = x 2 x 2 + a 2 − a 2 2 log ⁡ | x + x 2 + a 2 | {\displaystyle \int {\frac {x^{2}\;\mathrm {d} x}{\sqrt {x^{2}+a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\,\mathrm {arsinh} {\frac {x}{a}}={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\log \left|x+{\sqrt {x^{2}+a^{2}}}\right|}
∫ d x x x 2 + a 2 = − 1 a a r s i n h a x = − 1 a log ⁡ | a + x 2 + a 2 x | {\displaystyle \int {\frac {\mathrm {d} x}{x{\sqrt {x^{2}+a^{2}}}}}=-{\frac {1}{a}}\,\mathrm {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\log \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|}
∫ x 2 − a 2 d x = 1 2 ( x x 2 − a 2 ∓ a 2 a r c o s h | x a | ) (per  | x | ≥ | a | ;  −  per  x > 0 ,  +  per  x < 0 ) {\displaystyle \int {\sqrt {x^{2}-a^{2}}}\;\mathrm {d} x={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}\mp a^{2}\,\mathrm {arcosh} \left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(per }}|x|\geq |a|{\mbox{; }}-{\mbox{ per }}x>0{\mbox{, }}+{\mbox{ per }}x<0{\mbox{)}}}
∫ x x 2 − a 2 d x = 1 3 ( x 2 − a 2 ) 3 (per  | x | ≥ | a | ) {\displaystyle \int x{\sqrt {x^{2}-a^{2}}}\;\mathrm {d} x={\frac {1}{3}}{\sqrt {(x^{2}-a^{2})^{3}}}\qquad {\mbox{(per }}|x|\geq |a|{\mbox{)}}}
∫ x 2 − a 2 d x x = x 2 − a 2 − a arcsin ⁡ a x (per  | x | ≥ | a | ) {\displaystyle \int {\frac {{\sqrt {x^{2}-a^{2}}}\;\mathrm {d} x}{x}}={\sqrt {x^{2}-a^{2}}}-a\arcsin {\frac {a}{x}}\qquad {\mbox{(per }}|x|\geq |a|{\mbox{)}}}
∫ d x x 2 − a 2 = a r c o s h x a = log ⁡ ( | x | + x 2 − a 2 ) (per  | x | > | a | ) {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {x^{2}-a^{2}}}}=\mathrm {arcosh} {\frac {x}{a}}=\log \left(|x|+{\sqrt {x^{2}-a^{2}}}\right)\qquad {\mbox{(per }}|x|>|a|{\mbox{)}}}
∫ x d x x 2 − a 2 = x 2 − a 2 (per  | x | > | a | ) {\displaystyle \int {\frac {x\;\mathrm {d} x}{\sqrt {x^{2}-a^{2}}}}={\sqrt {x^{2}-a^{2}}}\qquad {\mbox{(per }}|x|>|a|{\mbox{)}}}
∫ x 2 d x x 2 − a 2 = x 2 x 2 − a 2 + a 2 2 a r c o s h | x a | = 1 2 ( x x 2 − a 2 + a 2 ln ⁡ ( | x | + x 2 − a 2 ) ) (per  | x | > | a | ) {\displaystyle \int {\frac {x^{2}\,\mathrm {d} x}{\sqrt {x^{2}-a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\,\mathrm {arcosh} \left|{\frac {x}{a}}\right|={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}+a^{2}\ln \left(|x|+{\sqrt {x^{2}-a^{2}}}\right)\right)\qquad {\mbox{(per }}|x|>|a|{\mbox{)}}}
∫ d x a x 2 + b x + c = 1 a ln ⁡ | 2 a ( a x 2 + b x + c ) + 2 a x + b | (per  a > 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a(ax^{2}+bx+c)}}+2ax+b\right|\qquad {\mbox{(per }}a>0{\mbox{)}}}
∫ d x a x 2 + b x + c = 1 a a r s i n h 2 a x + b 4 a c − b 2 (per  a > 0 ,  4 a c − b 2 > 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\mathrm {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(per }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}
∫ d x a x 2 + b x + c = 1 a log ⁡ | 2 a x + b | (per  a > 0 ,  4 a c − b 2 = 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\log |2ax+b|\qquad {\mbox{(per }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}
∫ d x a x 2 + b x + c = − 1 − a arcsin ⁡ 2 a x + b b 2 − 4 a c (per  a < 0 ,  4 a c − b 2 < 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(per }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}}
∫ x d x a x 2 + b x + c = a x 2 + b x + c a − b 2 a ∫ d x a x 2 + b x + c {\displaystyle \int {\frac {x\;\mathrm {d} x}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {ax^{2}+bx+c}}{a}}-{\frac {b}{2a}}\int {\frac {\mathrm {d} x}{\sqrt {ax^{2}+bx+c}}}}

Bibliografia

  • Murray R. Spiegel, Manuale di matematica, Etas Libri, 1974, pp. 61-73.
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