로그함수 적분표

아래 목록은 로그 함수의 부정적분이다. 아래의 적분식들에서 x > 0이며, 적분상수는 생략하였다. ∫ log a ⁡ x d x = x log a ⁡ x − x ln ⁡ a = x ln ⁡ x − x ln ⁡ a {\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}={\frac {x\ln x-x}{\ln a}}} ∫ ln ⁡ ( a x ) d x = x ln ⁡ ( a x ) − x {\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x} ∫ ln ⁡ ( a x + b ) d x = ( a x + b ) ln ⁡ ( a x + b ) − ( a x + b ) a {\displaystyle \int \ln(ax+b)\,dx={\frac {(ax+b)\ln(ax+b)-(ax+b)}{a}}} ∫ ( 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 ∑ k = 0 n ( − 1 ) n − k n ! k ! ( ln ⁡ x ) k {\displaystyle \int (\ln x)^{n}\,dx=x\sum _{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}}(\ln x)^{k}} ∫ d x ln ⁡ x = ln ⁡ | ln ⁡ x | + ln ⁡ x + ∑ k = 2 ∞ ( ln ⁡ x ) k k ⋅ k ! {\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}} ∫ d x ln ⁡ x = li ⁡ ( x ) {\displaystyle \int {\frac {dx}{\ln x}}=\operatorname {li} (x)} ( li ⁡ ( x ) {\displaystyle \operatorname {li} (x)} 는 로그 적분 함수) ∫ 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 {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{(}}n\neq 1{\mbox{)}}} ∫ ln ⁡ f ( x ) d x = x ln ⁡ f ( x ) − ∫ x f ′ ( x ) f ( x ) d x ( f 가 미분가능하고  f ( x ) > 0 일 때) {\displaystyle \int \ln f(x)\,dx=x\ln f(x)-\int x{\frac {f'(x)}{f(x)}}\,dx\qquad {\mbox{(}}f{\mbox{가 미분가능하고 }}f(x)>0{\mbox{일 때)}}} ∫ x m ln ⁡ x d x = x m + 1 ( ln ⁡ x m + 1 − 1 ( m + 1 ) 2 ) ( 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{(}}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 ( m ≠ − 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{(}}m\neq -1{\mbox{)}}} ∫ ( ln ⁡ x ) n d x x = ( ln ⁡ x ) n + 1 n + 1 ( n ≠ − 1 ) {\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(}}n\neq -1{\mbox{)}}} ∫ ln ⁡ x d x x m = − ln ⁡ x ( m − 1 ) x m − 1 − 1 ( m − 1 ) 2 x m − 1 ( 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{(}}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 ( m ≠ 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{(}}m\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 ( 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{(}}n\neq 1{\mbox{)}}} ∫ d x x ln ⁡ x = ln ⁡ | ln ⁡ x | {\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|} ∫ d x x ln ⁡ x ln ⁡ ln ⁡ x = ln ⁡ | ln ⁡ | ln ⁡ x | | {\displaystyle \int {\frac {dx}{x\ln x\ln \ln x}}=\ln \left|\ln \left|\ln x\right|\right|} ∫ d x x ln ⁡ ln ⁡ x = li ⁡ ( ln ⁡ x ) {\displaystyle \int {\frac {dx}{x\ln \ln x}}=\operatorname {li} (\ln x)} ∫ d x x n ln ⁡ x = ln ⁡ | ln ⁡ x | + ∑ k = 1 ∞ ( − 1 ) k ( n − 1 ) k ( ln ⁡ x ) k k ⋅ k ! {\displaystyle \int {\frac {dx}{x^{n}\ln x}}=\ln \left|\ln x\right|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(n-1)^{k}(\ln x)^{k}}{k\cdot k!}}} ∫ d x x ( ln ⁡ x ) n = − 1 ( n − 1 ) ( ln ⁡ x ) n − 1 ( n ≠ 1 ) {\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}} ∫ ln ⁡ ( x 2 + a 2 ) d x = x ln ⁡ ( x 2 + a 2 ) − 2 x + 2 a tan − 1 ⁡ x a {\displaystyle \int \ln(x^{2}+a^{2})\,dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}} ∫ x x 2 + a 2 ln ⁡ ( x 2 + a 2 ) d x = 1 4 ln 2 ⁡ ( x 2 + a 2 ) {\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})} ∫ 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))} ∫ 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)\,dx=e^{x}(x\ln x-x-\ln x)} ∫ 1 e x ( 1 x − ln ⁡ x ) d x = ln ⁡ x e x {\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\,dx={\frac {\ln x}{e^{x}}}} ∫ e x ( 1 ln ⁡ x − 1 x ( ln ⁡ x ) 2 ) d x = e x ln ⁡ x {\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x(\ln x)^{2}}}\right)\,dx={\frac {e^{x}}{\ln x}}} 로그함수를 n번 연속적으로 적분하는 경우, ∫ ln ⁡ x d x = x ( ln ⁡ x − 1 ) + C 0 {\displaystyle \int \ln x\,dx=x(\ln x-1)+C_{0}} 를 아래와 같이 일반화할 수 있다. ∫ ⋯ ∫ ln ⁡ x d x ⋯ d x = x n n ! ( ln x − ∑ k = 1 n 1 k ) + ∑ k = 0 n − 1 C k x k k ! {\displaystyle \int \dotsi \int \ln x\,dx\dotsm dx={\frac {x^{n}}{n!}}\left(\ln \,x-\sum _{k=1}^{n}{\frac {1}{k}}\right)+\sum _{k=0}^{n-1}C_{k}{\frac {x^{k}}{k!}}} • 적분표 • 로그 함수 Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. A few integrals are listed on page 69.