The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
Integrals involving only logarithmic functions
∫ log a x d x = x log a x − x ln a = x ln a ( ln x − 1 ) {\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}={\frac {x}{\ln a}}(\ln x-1)}
∫ ln ( a x ) d x = x ln ( a x ) − x = x ( ln ( a x ) − 1 ) {\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x=x(\ln(ax)-1)}
∫ ln ( a x + b ) d x = a x + b a ( ln ( a x + b ) − 1 ) {\displaystyle \int \ln(ax+b)\,dx={\frac {ax+b}{a}}(\ln(ax+b)-1)}
∫ ( 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 n x d x = ( − 1 ) n x ∑ k = 0 n n ! k ! ( − ln x ) k {\displaystyle \int {\ln ^{n}{x}\,dx}=(-1)^{n}\,x\,\sum _{k=0}^{n}{\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)} , the logarithmic integral.
∫ d x ( ln x ) n = − x ( n − 1 ) ( ln x ) n − 1 + 1 n − 1 ∫ d x ( ln x ) n − 1 (for 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{(for }}n\neq 1{\mbox{)}}}
∫ ln f ( x ) d x = x ln f ( x ) − ∫ x f ′ ( x ) f ( x ) d x (for differentiable f ( x ) > 0 ) {\displaystyle \int \ln f(x)\,dx=x\ln f(x)-\int x{\frac {f'(x)}{f(x)}}\,dx\qquad {\mbox{(for differentiable }}f(x)>0{\mbox{)}}}
Integrals involving logarithmic and power functions
∫ x m ln x d x = x m + 1 ( ln x m + 1 − 1 ( m + 1 ) 2 ) (for 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{(for }}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 (for 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{(for }}m\neq -1{\mbox{)}}}
∫ ( ln x ) n d x x = ( ln x ) n + 1 n + 1 (for n ≠ − 1 ) {\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
∫ ln x d x x m = − ln x ( m − 1 ) x m − 1 − 1 ( m − 1 ) 2 x m − 1 (for 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{(for }}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 (for 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{(for }}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 (for 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{(for }}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|} , etc.
∫ 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 (for n ≠ 1 ) {\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(for }}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})}
Integrals involving logarithmic and trigonometric functions
∫ 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))}
Integrals involving logarithmic and exponential functions
∫ 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}}}
n consecutive integrations For n {\displaystyle n} consecutive integrations, the formula
∫ ln x d x = x ( ln x − 1 ) + C 0 {\displaystyle \int \ln x\,dx=x(\ln x-1)+C_{0}}
generalizes to
∫ ⋯ ∫ 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!}}} , where C k {\displaystyle C_{k}} are arbitrary constants of integration.
See also List of mathematical identities Lists of mathematics topics
References
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.