Lista de integrais de funcións logarítmicas

A seguinte é unha lista de integrais de funcións logarítmicas.
Nota: neste artigo asúmese que x>0. ∫ ln ⁡ x d x
x ln ⁡ x − x {\displaystyle \int \ln x\, dx
x\ln x-x} ∫ ( ln ⁡ x ) 2 d x
x ( ln ⁡ x ) 2 − 2 x ln ⁡ x + 2 x {\displaystyle \int (\ln x)^{2}\; dx
x(\ln x)^{2}-2x\ln x+2x} ∫ ( ln ⁡ x ) n d x
x ( ln ⁡ x ) n − n ∫ ( ln ⁡ x ) n − 1 d x (para n ≠ 1 ) {\displaystyle \int (\ln x)^{n}\; dx
x(\ln x)^{n}-n\int (\ln x)^{n-1}dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ d x ln ⁡ x
ln ⁡ | ln ⁡ x | + ln ⁡ x + ∑ i
2 ∞ ( ln ⁡ x ) i i ⋅ i! {\displaystyle \int {\frac {dx}{\ln x}}
\ln |\ln x|+\ln x+\sum _{i
2}^{\infty }{\frac {(\ln x)^{i}}{i\cdot i! }}} ∫ d x ( ln ⁡ x ) n
− x ( n − 1 ) ( ln ⁡ x ) n − 1 + 1 n − 1 ∫ d x ( ln ⁡ x ) n − 1 (para n ≠ 1 ) {\displaystyle \int {\frac {dx}{(\ln x)^{n}}}
-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ x m ln ⁡ x d x
x m + 1 ( ln ⁡ x m + 1 − 1 ( m + 1 ) 2 ) (para m ≠ 1 ) {\displaystyle \int x^{m}\ln x\; dx
x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad {\mbox{(para }}m\neq 1{\mbox{)}}} ∫ x m ( ln ⁡ x ) n d x
x m + 1 ( ln ⁡ x ) n m + 1 − n m + 1 ∫ x m ( ln ⁡ x ) n − 1 d x (para m, n ≠ 1 ) {\displaystyle \int x^{m}(\ln x)^{n}\; dx
{\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\qquad {\mbox{(para }}m, n\neq 1{\mbox{)}}} ∫ ( ln ⁡ x ) n d x x
( ln ⁡ x ) n + 1 n + 1 (para n ≠ 1 ) {\displaystyle \int {\frac {(\ln x)^{n}\; dx}{x}}
{\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ ln ⁡ x d x x m
− ln ⁡ x ( m − 1 ) x m − 1 − 1 ( m − 1 ) 2 x m − 1 (para m ≠ 1 ) {\displaystyle \int {\frac {\ln x\, dx}{x^{m}}}
-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}\qquad {\mbox{(para }}m\neq 1{\mbox{)}}} ∫ ( ln ⁡ x ) n d x x m
− ( ln ⁡ x ) n ( m − 1 ) x m − 1 + n m − 1 ∫ ( ln ⁡ x ) n − 1 d x x m (para m, n ≠ 1 ) {\displaystyle \int {\frac {(\ln x)^{n}\; dx}{x^{m}}}
-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\qquad {\mbox{(para }}m, n\neq 1{\mbox{)}}} ∫ x m d x ( ln ⁡ x ) n
− x m + 1 ( n − 1 ) ( ln ⁡ x ) n − 1 + m + 1 n − 1 ∫ x m d x ( ln ⁡ x ) n − 1 (para n ≠ 1 ) {\displaystyle \int {\frac {x^{m}\; dx}{(\ln x)^{n}}}
-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ d x x ln ⁡ x
ln ⁡ | ln ⁡ x | {\displaystyle \int {\frac {dx}{x\ln x}}
\ln |\ln x|} ∫ d x x n ln ⁡ x
ln ⁡ | ln ⁡ x | + ∑ i
1 ∞ ( − 1 ) i ( n − 1 ) i ( ln ⁡ x ) i i ⋅ i! {\displaystyle \int {\frac {dx}{x^{n}\ln x}}
\ln |\ln x|+\sum _{i
1}^{\infty }(-1)^{i}{\frac {(n-1)^{i}(\ln x)^{i}}{i\cdot i! }}} ∫ d x x ( ln ⁡ x ) n
− 1 ( n − 1 ) ( ln ⁡ x ) n − 1 (para n ≠ 1 ) {\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}
-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ sin ⁡ ( ln ⁡ x ) d x
x 2 ( sin ⁡ ( ln ⁡ x ) − cos ⁡ ( ln ⁡ x ) ) {\displaystyle \int \sin(\ln x)\; dx
{\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))} ∫ cos ⁡ ( ln ⁡ x ) d x
x 2 ( sin ⁡ ( ln ⁡ x ) + cos ⁡ ( ln ⁡ x ) ) {\displaystyle \int \cos(\ln x)\; dx
{\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}