Ondorengoa funtzio arrazionalen integralen zerrenda bat da (jatorrizkoak edo antideribatuak). Integralen zerrenda osatuago nahi baduzu, ikusi integralen zerrenda.
![{\displaystyle \int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+K\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e57d567980c1bb510b2e468fa318925d3b91534d)
![{\displaystyle \int {\frac {c}{ax+b}}dx={\frac {c}{a}}\ln \left|ax+b\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99aec6dfc06956c705c007f5e2e60de25ba3be5f)
![{\displaystyle \int x(ax+b)^{n}dx={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}+K\qquad {\mbox{( }}n\not \in \{-1,-2\}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4927cb20218a5369df3ba40f29b5bcd1e642d2f)
![{\displaystyle \int {\frac {x}{ax+b}}dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5c87785a689f5df1d852a7b43343b994f6444b)
![{\displaystyle \int {\frac {x}{(ax+b)^{2}}}dx={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3e8dbd4ecca039bb7e69e5e310aa169065fbe7)
![{\displaystyle \int {\frac {x}{(ax+b)^{n}}}dx={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}+K\qquad {\mbox{( }}n\not \in \{1,2\}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b526d8928744ef77e674092bd27a1d125dd3121c)
![{\displaystyle \int {\frac {f'(x)}{f(x)}}dx=\ln \left|f(x)\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5290cc093abee85b86b96b07aaa2c919f54f1a)
![{\displaystyle \int {\frac {x^{2}}{ax+b}}dx={\frac {b^{2}\ln(\left|ax+b\right|)}{a^{3}}}+{\frac {ax^{2}-2bx}{2a^{2}}}+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cef907189cac9fc5a59c082b920a072863977f57)
![{\displaystyle \int {\frac {x^{2}}{(ax+b)^{2}}}dx={\frac {1}{a^{3}}}\left(ax-2b\ln \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90a4f9ba4dda4a69353463f53106aa46ba3a2a99)
![{\displaystyle \int {\frac {x^{2}}{(ax+b)^{3}}}dx={\frac {1}{a^{3}}}\left(\ln \left|ax+b\right|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right)+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b00fb6d6c15873ac78be4cdf7d9e6d3b1a31e22)
![{\displaystyle \int {\frac {x^{2}}{(ax+b)^{n}}}dx={\frac {1}{a^{3}}}\left(-{\frac {(ax+b)^{3-n}}{(n-3)}}+{\frac {2b(ax+b)^{2-n}}{(n-2)}}-{\frac {b^{2}(ax+b)^{1-n}}{(n-1)}}\right)+K\qquad {\mbox{( }}n\not \in \{1,2,3\}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1422cc019280399651ef09b69656a7715cef4880)
![{\displaystyle \int {\frac {1}{x(ax+b)}}dx=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca9cd1748dd860a9b08a72a1c9a8885e2fa5cc7)
![{\displaystyle \int {\frac {1}{x^{2}(ax+b)}}dx=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e2927b166025f8921c1e3458cd478431b87883)
![{\displaystyle \int {\frac {1}{x^{2}(ax+b)^{2}}}dx=-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)+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f61107e48ca8786419bcb37730baebce338369b)
![{\displaystyle \int {\frac {1}{x^{2}+a^{2}}}dx={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f5e4f0f6c2f81c0ce5cd06e3e906632b5a7994)
![{\displaystyle \int {\frac {1}{x^{2}-a^{2}}}dx={\begin{cases}\displaystyle -{\frac {1}{a}}\,\mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}+K&{\mbox{(for }}|x|<|a|{\mbox{)}}\\[12pt]\displaystyle -{\frac {1}{a}}\,\mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}+K&{\mbox{( }}|x|>|a|{\mbox{)}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4211b35a9535ac4ec88df4abdfccc1ccce7f81a)
bada:
![{\displaystyle \int {\frac {1}{ax^{2}+bx+c}}dx={\begin{cases}\displaystyle {\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}+K&{\mbox{( }}4ac-b^{2}>0{\mbox{)}}\\[12pt]\displaystyle -{\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+K={\frac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|+K&{\mbox{( }}4ac-b^{2}<0{\mbox{)}}\\[12pt]\displaystyle -{\frac {2}{2ax+b}}+K&{\mbox{( }}4ac-b^{2}=0{\mbox{)}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2863cf805a50629ae0466988426ee30e7cfaaf)
![{\displaystyle \int {\frac {x}{ax^{2}+bx+c}}dx={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {dx}{ax^{2}+bx+c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/742ca6fd0f058dcd9e83be78b566a387df2ad396)
![{\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\begin{cases}\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}}}}+K&{\mbox{( }}4ac-b^{2}>0{\mbox{)}}\\[12pt]\displaystyle {\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+K&{\mbox{( }}4ac-b^{2}<0{\mbox{)}}\\[12pt]\displaystyle {\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}+K&{\mbox{( }}4ac-b^{2}=0{\mbox{)}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dbff163b1e8ad2bc4b3f41e785c7c38af074502)
![{\displaystyle \int {\frac {1}{(ax^{2}+bx+c)^{n}}}dx={\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}}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f442c8210ff813e09db05ab613b244917c821166)
![{\displaystyle \int {\frac {x}{(ax^{2}+bx+c)^{n}}}dx=-{\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}}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c731cb1b9e4e05abd26f9c9f66d70ae3c18a608)
![{\displaystyle \int {\frac {1}{x(ax^{2}+bx+c)}}dx={\frac {1}{2c}}\ln \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {1}{ax^{2}+bx+c}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505fd268d136f336322d8635315c3983c3c8aba6)
![{\displaystyle \int {\frac {dx}{x^{2^{n}}+1}}=\sum _{k=1}^{2^{n-1}}\left\{{\frac {1}{2^{n-1}}}\left[\sin \left({\frac {(2k-1)\pi }{2^{n}}}\right)\arctan \left[\left(x-\cos \left({\frac {(2k-1)\pi }{2^{n}}}\right)\right)\csc \left({\frac {(2k-1)\pi }{2^{n}}}\right)\right]\right]-{\frac {1}{2^{n}}}\left[\cos \left({\frac {(2k-1)\pi }{2^{n}}}\right)\ln \left|x^{2}-2x\cos \left({\frac {(2k-1)\pi }{2^{n}}}\right)+1\right|\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/493e98d39eebbb6342aa5c928ba6da54755b63fe)
Edozein funtzio arrazional integra dezakegu goiko berdintzak erabiliz eta zatiki arrazionalen integrazioaren artikuluan eskaintzen diren teknikak aplikatuz, funtzio arrazionalak ondorengo formako batugaietan banatzearen bidez:
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