List of integrals of logarithmic functions

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 ⁡ x ) n d x = ( − 1 ) n n ! x ∑ k = 0 n ( − ln ⁡ x ) k k ! {\displaystyle \int (\ln x)^{n}\,dx=(-1)^{n}n!x\sum _{k=0}^{n}{\frac {(-\ln x)^{k}}{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!}}}

See also

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.