Ovo je spisak integrala (antiderivativnih funkcija) racionalnih funkcija. Za kompletniji spisak integrala, pogledajte tabelu integrala i spisak integrala.
![{\displaystyle \int (ax+b)^{n}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de3a5696d6a0706c62c9fecfd55b741f8d75fa09) |
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![{\displaystyle \int {\frac {dx}{ax+b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9bc5536cf14b5f472b90597c6c3a1333751d18) |
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![{\displaystyle \int x(ax+b)^{n}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/333036d5f19b0939533abfa39bd786fa4dd52b14) |
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![{\displaystyle \int {\frac {xdx}{ax+b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1bec455345a8dc4f1a69fda6f0e0c6e637c739) |
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![{\displaystyle \int {\frac {xdx}{(ax+b)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/395b0ec6b36f9775e0e47bb33f66521ec45af4ab) |
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![{\displaystyle \int {\frac {xdx}{(ax+b)^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df5fa7422fea4b5acc0bae359df9b19b33f76af6) |
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![{\displaystyle \int {\frac {x^{2}dx}{ax+b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d75e25483485d53ebacd4047563af824fac7acd) |
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![{\displaystyle \int {\frac {x^{2}dx}{(ax+b)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79a1cb9e261c0b0880f42ec1a2d65587d5155455) |
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![{\displaystyle \int {\frac {x^{2}dx}{(ax+b)^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1fc9a850fae1e4db6a2d5f71d0606c28409be4) |
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![{\displaystyle \int {\frac {x^{2}dx}{(ax+b)^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bfa615b8536e79b59d80a5677fbc9405557e69b) |
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![{\displaystyle \int {\frac {dx}{x(ax+b)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/834e3044f95e6242735ebc84703a23d634bf431d) |
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![{\displaystyle \int {\frac {dx}{x^{2}(ax+b)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d25e2a6aca52d9cc10331fe9638dcd4041626805) |
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![{\displaystyle \int {\frac {dx}{x^{2}(ax+b)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dce2864f6cabb840918e87c48bad16cda0f3689) |
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![{\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f18d08c1262103b3e95bea0bb0b813cc3c88d85) |
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![{\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b29b01e6f28dec471bd26480d186d3d2ec25f2) |
![{\displaystyle -{\frac {1}{a}}\,\mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}\qquad {\mbox{(za }}|x|<|a|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4112980ec286ed17357d891b56bbf212902e6897)
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![{\displaystyle -{\frac {1}{a}}\,\mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}\qquad {\mbox{(za }}|x|>|a|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9b13deb63445476c01d75300f0daeca6c67314)
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![{\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a78db8cfaf7712406438d024bed5ce723dea53e) |
![{\displaystyle {\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(za }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b38822559dac4570b14fa038c02c5b4fd0d19d)
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![{\displaystyle -{\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|\qquad {\mbox{(za }}4ac-b^{2}<0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf9158fa99096b79ec8c857f74f90dcdbc69e7b)
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![{\displaystyle -{\frac {2}{2ax+b}}\qquad {\mbox{(za }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2a277eb89d078a8352d64bc95d00e75db6b094)
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![{\displaystyle \int {\frac {xdx}{ax^{2}+bx+c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a521edd5eafe2e4a523855542756710edf160388) |
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![{\displaystyle \int {\frac {(mx+n)dx}{ax^{2}+bx+c}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d03fd48a7673558ab47c6652f7a17f03fc6950) |
![{\displaystyle {\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}}}}\qquad {\mbox{(za }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e478408761b5339b6f39f51ffd12a34fb394355f)
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![{\displaystyle {\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(za }}4ac-b^{2}<0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae7c566269b54f8dbaebff97c82fbf9ad9a5a4f)
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![{\displaystyle {\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}\,\,\,\,\,\,\,\,\,\,\qquad {\mbox{(za }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/356f414424cafdbf4a014ad2000b1ca5366c1227)
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![{\displaystyle \int {\frac {dx}{(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 {dx}{(ax^{2}+bx+c)^{n-1}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c079949856bc293bcbdbe3de566cf9359a42a3)
![{\displaystyle \int {\frac {xdx}{(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 {dx}{(ax^{2}+bx+c)^{n-1}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b280c9b89ceb73d3d9f522672aa9639e786a4b6)
![{\displaystyle \int {\frac {dx}{x(ax^{2}+bx+c)}}={\frac {1}{2c}}\ln \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {dx}{ax^{2}+bx+c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c358c1f184af0f3b79eb1e30dd5bb94ca47304)
Bilo koja racionalna funkcija se može integrisati koristeći gore navedene jednačine i parcijalne razlomke u integraciji, rastavljajući racionalnu funkciju na sumu razlomaka po formi:
![{\displaystyle {\frac {ex+f}{\left(ax^{2}+bx+c\right)^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/785692bdf4082452ab13d9424805cbcab7381cc6)
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