对数函数积分表

以下是部份对数函數的積分表(書寫時省略了不定積分結果中都含有的任意常數Cn)。 注：我们规定 { log α 2 ⁡ x = log α ⁡ log α ⁡ x log α 2 ⁡ | x | = log α ⁡ | ( log α ⁡ | x | ) | {\displaystyle {\begin{cases}\log _{\alpha }^{2}x=\log _{\alpha }\log _{\alpha }x\\\log _{\alpha }^{2}|x|=\log _{\alpha }|(\log _{\alpha }|x|)|\end{cases}}} { ∫ ln x   d x = x ln x − ∫ x 1 x   d x = x ln x − ∫ d x = x ln x − x ∫ ln ⁡ ( − x )   d x = − ∫ ln ⁡ ( − x )   d ( − x ) = − [ ( − x ) ln ⁡ ( − x ) − ( − x ) ] = x ln ⁡ ( − x ) − x ⟹ { ∫ ln | x | = x ln | x | − x ∫ log α | x | d x = ∫ ln | x | ln α d x = x ln | x | − x ln α = x log α | x | − x ln α {\displaystyle {\begin{cases}\int \ln \!x\ {\mbox{d}}x&=x\ln \!x-\int x\!{\dfrac {1}{x}}\ {\mbox{d}}x\\&=x\ln \!x-\int \!\!{\mbox{d}}x\\&=x\ln \!x-x\\\int \ln(-x)\ {\mbox{d}}x&=-\int \ln(-x)\ {\mbox{d}}\!(-x)\\&=-\left[\left(-x\right)\ln \left(-x\right)-\left(-x\right)\right]\\&=x\ln \left(-x\right)-x\end{cases}}\Longrightarrow {\begin{cases}\int \ln \!|x|&=x\ln \!|x|-x\\\int \log _{\alpha }\!|x|{\mbox{d}}x&=\int {\dfrac {\ln \!|x|}{\ln \!\alpha }}{\mbox{d}}x\\&={\dfrac {x\ln \!|x|-x}{\ln \!\alpha }}\\&=x\log _{\alpha }\!|x|-{\dfrac {x}{\ln \!\alpha }}\end{cases}}} ∫ ln M x d x = x ln M x − x {\displaystyle \int \ln \!Mx{\mbox{d}}x=x\ln \!Mx-x} ∫ ( ln x ) 2 d x = x ( ln x ) 2 − 2 x ln x + 2 x {\displaystyle \int (\ln \!x)^{2}{\mbox{d}}x=x(\ln \!x)^{2}-2x\ln \!x+2x} ∫ ( ln M x ) n d x = x ( ln M x ) n − n ∫ ( ln M x ) n − 1 d x {\displaystyle \int (\ln \!Mx)^{n}{\mbox{d}}x=x(\ln \!Mx)^{n}-n\int (\ln \!Mx)^{n-1}dx} ∫ d x ln x = ln | ln x | + ln x + ∑ I = 2 ∞ ( ln x ) I I ! I {\displaystyle \int {\frac {{\mbox{d}}x}{\ln \!x}}=\ln \!|\!\ln \!x|+\ln \!x+\sum _{I=2}^{\infty }{\frac {(\ln \!x)^{I}}{I!I}}} ∫ d x ( ln x ) n = − x ( n − 1 ) ( ln x ) n − 1 + 1 n − 1 ∫ d x ( ln ⁡ x ) n − 1 ( n ≠ 1 ) {\displaystyle \int {\frac {{\mbox{d}}x}{(\ln \!x)^{n}}}=-{\frac {x}{(n-1)(\ln \!x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {{\mbox{d}}x}{(\ln x)^{n-1}}}\qquad (n\neq 1)} ∫ x N ln x d x = x N + 1 ( ln x N + 1 − 1 ( N + 1 ) 2 ) ( N ≠ − 1 ) {\displaystyle \int x^{N}\ln \!x{\mbox{d}}x=x^{N+1}\left({\frac {\ln \!x}{N+1}}-{\frac {1}{(N+1)^{2}}}\right)\qquad (N\neq -1)} ∫ x N ( ln x ) n   d x = x N + 1 ( ln x ) n N + 1 − n N + 1 ∫ x N ( ln ⁡ x ) n − 1 d x ( N ≠ − 1 ) {\displaystyle \int x^{N}(\ln \!x)^{n}\ {\mbox{d}}x={\frac {x^{N+1}(\ln \!x)^{n}}{N+1}}-{\frac {n}{N+1}}\int x^{N}(\ln x)^{n-1}dx\qquad (N\neq -1)} ∫ ( ln x ) n   d x x = ( ln x ) n + 1 n + 1 ( n ≠ − 1 ) {\displaystyle \int {\frac {(\ln \!x)^{n}\ {\mbox{d}}x}{x}}={\frac {(\ln \!x)^{n+1}}{n+1}}\qquad (n\neq -1)} ∫ ln x   d x x r = − ln x ( r − 1 ) x r − 1 − 1 ( 1 − r ) 2 x r − 1 ( r ≠ 1 ) {\displaystyle \int {\frac {\ln \!x\ {\mbox{d}}x}{x^{r}}}=-{\frac {\ln \!x}{(r-1)x^{r-1}}}-{\frac {1}{(1-r)^{2}x^{r-1}}}\qquad (r\neq 1)} ∫ ( ln x ) n   d x x r = − ( ln ⁡ x ) n ( r − 1 ) x r − 1 + n r − 1 ∫ ( ln ⁡ x ) n − 1 d x x m ( m ≠ 1 ) {\displaystyle \int {\frac {(\ln \!x)^{n}\ {\mbox{d}}x}{x^{r}}}=-{\frac {(\ln x)^{n}}{(r-1)x^{r-1}}}+{\frac {n}{r-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\qquad (m\neq 1)} ∫ x N   d x ( ln x ) n = − x N + 1 ( n − 1 ) ( ln x ) n − 1 + N + 1 n − 1 ∫ x N d x ( ln x ) n − 1 ( n ≠ 1 ) {\displaystyle \int {\frac {x^{N}\ {\mbox{d}}x}{(\ln \!x)^{n}}}=-{\frac {x^{N+1}}{(n-1)(\ln \!x)^{n-1}}}+{\frac {N+1}{n-1}}\int {\frac {x^{N}{\mbox{d}}x}{(\ln \!x)^{n-1}}}\qquad (n\neq 1)} ∫ d x x ln x = ln | ln x | {\displaystyle \int {\frac {{\mbox{d}}x}{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 {{\mbox{d}}x}{x^{n}\ln \!x}}=\ln \!|\!\ln \!x|+\sum _{I=1}^{\infty }(-1)^{I}{\frac {(n-1)^{I}(\ln \!x)^{I}}{I!I}}} ∫ d x x ( ln x ) n = 1 ( 1 − n ) ( ln x ) n − 1 ( n ≠ 1 ) {\displaystyle \int {\frac {{\mbox{d}}x}{x(\ln \!x)^{n}}}={\frac {1}{(1-n)(\ln \!x)^{n-1}}}\qquad (n\neq 1)} ∫ sin ⁡ ln x   d x = x ( sin ⁡ ln ⁡ x − cos ⁡ ln x ) 2 {\displaystyle \int \sin \ln \!x\ {\mbox{d}}x={\frac {x(\sin \ln x-\cos \ln \!x)}{2}}} ∫ cos ⁡ ln x   d x = x ( sin ⁡ ln ⁡ x + cos ⁡ ln x ) 2 {\displaystyle \int \cos \ln \!x\ {\mbox{d}}x={\frac {x(\sin \ln x+\cos \ln \!x)}{2}}} ∫ e x ( x ln x − x − 1 x )   d x = e x ( x ln x − x − ln x ) {\displaystyle \int e^{x}\left(x\ln \!x-x-{\frac {1}{x}}\right)\ {\mbox{d}}x=e^{x}(x\ln \!x-x-\ln \!x)}