Questa pagina contiene una tavola di integrali indefiniti di funzioni logaritmiche. Per altri integrali vedi Integrale § Tavole di integrali.
In questa pagina si assume che x sia una variabile sull'insieme dei reali positivi. C denota una costante arbitraria d'integrazione, specificabile solo per un valore particolare dell'integrale.
![{\displaystyle \int \log cx\,\mathrm {d} x=x\log cx-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/608475be51d5cd5a69c3f3a00948efcb5b5379a7)
![{\displaystyle \int (\log x)^{2}\;\mathrm {d} x=x(\log x)^{2}-2x\log x+2x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dce823d2ab291ca482b1ac8f334b07d4578efaa4)
![{\displaystyle \int (\log cx)^{n}\;\mathrm {d} x=x(\log cx)^{n}-n\int (\log cx)^{n-1}\,\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52595e7b9ae6d83b342b2c36422e734c783a37ca)
![{\displaystyle \int {\frac {\mathrm {d} x}{\log x}}=\log |\log x|+\log x+\sum _{k=2}^{\infty }{\frac {(\log x)^{k}}{k\cdot k!}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59edfc7cf7db7d42d24e6708708bf1700ef743fd)
![{\displaystyle \int {\frac {\mathrm {d} x}{(\log x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {\mathrm {d} x}{(\log x)^{n-1}}}+C\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be0b304659296e98e71687003e0793f4f7dba1b1)
![{\displaystyle \int x^{m}\log x\;\mathrm {d} x=x^{m+1}\left({\frac {\log x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)+C\qquad {\mbox{(per }}m\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1520ef210c5c91531cc4e4603bf6a8600d0eeb56)
![{\displaystyle \int x^{m}(\log x)^{n}\;\mathrm {d} x={\frac {x^{m+1}(\log x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\log x)^{n-1}\mathrm {d} x+C\qquad {\mbox{(per }}m,n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f224cdc1532f74c9793398ee685d1ff8cc55bb)
![{\displaystyle \int {\frac {(\log x)^{n}\;\mathrm {d} x}{x}}={\frac {(\log x)^{n+1}}{n+1}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1e173a5cc20ff834ea6bdd963268b9825d9e44)
![{\displaystyle \int {\frac {\log x\,\mathrm {d} x}{x^{m}}}=-{\frac {\log x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}+C\qquad {\mbox{(per }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8373e811ecf96c7bef54f84abed1a3a2e5cda8f)
![{\displaystyle \int {\frac {(\log x)^{n}\;\mathrm {d} x}{x^{m}}}=-{\frac {(\log x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\log x)^{n-1}\mathrm {d} x}{x^{m}}}+C\qquad {\mbox{(per }}m,n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd2dc56be4941fd8d1bc8be2eded59af20a6dd9)
![{\displaystyle \int {\frac {x^{m}\;\mathrm {d} x}{(\log x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\log x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}\mathrm {d} x}{(\log x)^{n-1}}}+C\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/213bc79c5607052a147e40e68c1c16077a305153)
![{\displaystyle \int {\frac {\mathrm {d} x}{x\log x}}=\log |\log x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63d9c4a2ea5cbad8fc757e16e67a3fdb6545fd19)
![{\displaystyle \int {\frac {\mathrm {d} x}{x^{n}\log x}}=\log |\log x|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(n-1)^{k}(\log x)^{k}}{k\cdot k!}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aad2f21bda5aced306837fae591eeefb0fb3dd)
![{\displaystyle \int {\frac {\mathrm {d} x}{x(\log x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}+C\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d662766a3b5dc8f2a13ad13472db32374c9eec4)
![{\displaystyle \int \sin(\log x)\;\mathrm {d} x={\frac {x}{2}}[\sin(\log x)-\cos(\log x)]+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e77bb0762276f2780afa176e9b02334307ac8302)
![{\displaystyle \int \cos(\log x)\;\mathrm {d} x={\frac {x}{2}}[\sin(\log x)+\cos(\log x)]+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c75f6a0558dce736b24d4fb7a0d898a442a51cc9)
Bibliografia
- Murray R. Spiegel, Manuale di matematica, Etas Libri, 1974, p. 86.
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