Questa pagina contiene una tavola di integrali indefiniti di funzioni irrazionali. Per altri integrali vedi Tavole di Integrali.
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef1f97080f9bf46828d314aea2bab6a344b8192)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54e57be9bf848fae43f73c3e257df21f1ce8e2ae)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/632b3813736c2b660179f54200f26a5ea577b545)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9a92a66536e1f70279c15a9903109f03db71ff)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb20bbb92accaf4ca776d90135d777878916ed4)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/975d51c0cd404d0b9d18e843e44b8b6f19f8d02e)
![{\displaystyle \int x{\sqrt {x^{2}+a^{2}}}\;\mathrm {d} x={\frac {1}{3}}{\sqrt {(x^{2}+a^{2})^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20e2789afaf51688d3e35f532e3e57bf61dc1f01)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad8d0f51644436f22975dc56bcbf96ce53a30129)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c85df20f79995bd5e68f72102b993568388f0e)
![{\displaystyle \int {\frac {x\,\mathrm {d} x}{\sqrt {x^{2}+a^{2}}}}={\sqrt {x^{2}+a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c3a458bffc1f7b5eced00e7589a005307fcf08)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8745420048b74e14761e92690f485a1e965aea63)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e75b718ea2af552df9cabb61d6d9b1810e89205)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0e2efb0789c36566ef259c533711e066d3e795)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/113ea81d5f25b155116e96cf0b7e8743d1476b99)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc04880e82722c77f7958f8dc4b0bc35463be77)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0937ca217537c210944933ac91c980378c8cfcf5)
![{\displaystyle \int {\frac {x\;\mathrm {d} x}{\sqrt {x^{2}-a^{2}}}}={\sqrt {x^{2}-a^{2}}}\qquad {\mbox{(per }}|x|>|a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f52ba932c2d8420b808db34d3bd1d35c9e7d28)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f45efd17cd95141c726670e7811f94b4470f4c7)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c41bdedd9029f52568046f3bb9dab7a9761fb8e4)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b465286a56c827e975f4a5bc5211a21bbd9c1be8)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a4c26cf09061d485e622446df58ad034a3aeea)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f95b6cacac279ab1a664f16f19e61248bfb7f707)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e49a1f4194615c053f7eb3b5e881047d30d1860a)
Bibliografia
- Murray R. Spiegel, Manuale di matematica, Etas Libri, 1974, pp. 61-73.
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