Tavola degli integrali indefiniti di funzioni logaritmiche

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.

∫ log ⁡ c x d x = x log ⁡ c x − x + C {\displaystyle \int \log cx\,\mathrm {d} x=x\log cx-x+C}
∫ ( log ⁡ x ) 2 d x = x ( log ⁡ x ) 2 − 2 x log ⁡ x + 2 x + C {\displaystyle \int (\log x)^{2}\;\mathrm {d} x=x(\log x)^{2}-2x\log x+2x+C}
∫ ( log ⁡ c x ) n d x = x ( log ⁡ c x ) n − n ∫ ( log ⁡ c x ) n − 1 d x (per  n ≠ 1 ) {\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{)}}}
∫ d x log ⁡ x = log ⁡ | log ⁡ x | + log ⁡ x + ∑ k = 2 ∞ ( log ⁡ x ) k k ⋅ k ! + C {\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}
∫ d x ( log ⁡ x ) n = − x ( n − 1 ) ( ln ⁡ x ) n − 1 + 1 n − 1 ∫ d x ( log ⁡ x ) n − 1 + C (per  n ≠ 1 ) {\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{)}}}
∫ x m log ⁡ x d x = x m + 1 ( log ⁡ x m + 1 − 1 ( m + 1 ) 2 ) + C (per  m ≠ − 1 ) {\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{)}}}
∫ x m ( log ⁡ x ) n d x = x m + 1 ( log ⁡ x ) n m + 1 − n m + 1 ∫ x m ( log ⁡ x ) n − 1 d x + C (per  m , n ≠ 1 ) {\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{)}}}
∫ ( log ⁡ x ) n d x x = ( log ⁡ x ) n + 1 n + 1 + C {\displaystyle \int {\frac {(\log x)^{n}\;\mathrm {d} x}{x}}={\frac {(\log x)^{n+1}}{n+1}}+C}
∫ log ⁡ x d x x m = − log ⁡ x ( m − 1 ) x m − 1 − 1 ( m − 1 ) 2 x m − 1 + C (per  m ≠ 1 ) {\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{)}}}
∫ ( log ⁡ x ) n d x x m = − ( log ⁡ x ) n ( m − 1 ) x m − 1 + n m − 1 ∫ ( log ⁡ x ) n − 1 d x x m + C (per  m , n ≠ 1 ) {\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{)}}}
∫ x m d x ( log ⁡ x ) n = − x m + 1 ( n − 1 ) ( log ⁡ x ) n − 1 + m + 1 n − 1 ∫ x m d x ( log ⁡ x ) n − 1 + C (per  n ≠ 1 ) {\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{)}}}
∫ d x x log ⁡ x = log ⁡ | log ⁡ x | + C {\displaystyle \int {\frac {\mathrm {d} x}{x\log x}}=\log |\log x|+C}
∫ d x x n log ⁡ x = log ⁡ | log ⁡ x | + ∑ k = 1 ∞ ( − 1 ) k ( n − 1 ) k ( log ⁡ x ) k k ⋅ k ! + C {\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}
∫ d x x ( log ⁡ x ) n = − 1 ( n − 1 ) ( ln ⁡ x ) n − 1 + C (per  n ≠ 1 ) {\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{)}}}
∫ sin ⁡ ( log ⁡ x ) d x = x 2 [ sin ⁡ ( log ⁡ x ) − cos ⁡ ( log ⁡ x ) ] + C {\displaystyle \int \sin(\log x)\;\mathrm {d} x={\frac {x}{2}}[\sin(\log x)-\cos(\log x)]+C}
∫ cos ⁡ ( log ⁡ x ) d x = x 2 [ sin ⁡ ( log ⁡ x ) + cos ⁡ ( log ⁡ x ) ] + C {\displaystyle \int \cos(\log x)\;\mathrm {d} x={\frac {x}{2}}[\sin(\log x)+\cos(\log x)]+C}

Bibliografia

  • Murray R. Spiegel, Manuale di matematica, Etas Libri, 1974, p. 86.
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