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List of integrals of logarithmic functions

From Wikipedia, the free encyclopedia

The following is a list ofintegrals (antiderivative functions) oflogarithmic functions. For a complete list of integral functions, seelist of integrals.

Note:x > 0 is assumed throughout this article, and theconstant of integration is omitted for simplicity.

Integrals involving only logarithmic functions

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∫loga⁡xdx=xloga⁡x−xln⁡a=xln⁡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⁡(ax)dx=xln⁡(ax)−x=x(ln⁡(ax)−1){\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x=x(\ln(ax)-1)}
∫ln⁡(ax+b)dx=ax+ba(ln⁡(ax+b)−1){\displaystyle \int \ln(ax+b)\,dx={\frac {ax+b}{a}}(\ln(ax+b)-1)}
∫(ln⁡x)2dx=x(ln⁡x)2−2xln⁡x+2x{\displaystyle \int (\ln x)^{2}\,dx=x(\ln x)^{2}-2x\ln x+2x}
∫lnn⁡xdx=(−1)nx∑k=0nn!k!(−ln⁡x)k{\displaystyle \int {\ln ^{n}{x}\,dx}=(-1)^{n}\,x\,\sum _{k=0}^{n}{\frac {n!}{k!}}(-\ln x)^{k}}
∫dxln⁡x=ln⁡|ln⁡x|+ln⁡x+∑k=2∞(ln⁡x)kk⋅k!{\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}
∫dxln⁡x=li⁡(x){\displaystyle \int {\frac {dx}{\ln x}}=\operatorname {li} (x)}, thelogarithmic integral.
∫dx(ln⁡x)n=−x(n−1)(ln⁡x)n−1+1n−1∫dx(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)dx=xln⁡f(x)−∫xf′(x)f(x)dx(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

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∫xmln⁡xdx=xm+1(ln⁡xm+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{)}}}
∫xm(ln⁡x)ndx=xm+1(ln⁡x)nm+1−nm+1∫xm(ln⁡x)n−1dx(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)ndxx=(ln⁡x)n+1n+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⁡xdxxm=−ln⁡x(m−1)xm−1−1(m−1)2xm−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)ndxxm=−(ln⁡x)n(m−1)xm−1+nm−1∫(ln⁡x)n−1dxxm(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{)}}}
∫xmdx(ln⁡x)n=−xm+1(n−1)(ln⁡x)n−1+m+1n−1∫xmdx(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{)}}}
∫dxxln⁡x=ln⁡|ln⁡x|{\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}
∫dxxln⁡xln⁡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.
∫dxxln⁡ln⁡x=li⁡(ln⁡x){\displaystyle \int {\frac {dx}{x\ln \ln x}}=\operatorname {li} (\ln x)}
∫dxxnln⁡x=ln⁡|ln⁡x|+∑k=1∞(−1)k(n−1)k(ln⁡x)kk⋅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!}}}
∫dxx(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⁡(x2+a2)dx=xln⁡(x2+a2)−2x+2atan−1⁡xa{\displaystyle \int \ln(x^{2}+a^{2})\,dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}
∫xx2+a2ln⁡(x2+a2)dx=14ln2⁡(x2+a2){\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

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∫sin⁡(ln⁡x)dx=x2(sin⁡(ln⁡x)−cos⁡(ln⁡x)){\displaystyle \int \sin(\ln x)\,dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}
∫cos⁡(ln⁡x)dx=x2(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

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∫ex(xln⁡x−x−1x)dx=ex(xln⁡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)}
∫1ex(1x−ln⁡x)dx=ln⁡xex{\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\,dx={\frac {\ln x}{e^{x}}}}
∫ex(1ln⁡x−1x(ln⁡x)2)dx=exln⁡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

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Forn{\displaystyle n} consecutive integrations, the formula

∫ln⁡xdx=x(ln⁡x−1)+C0{\displaystyle \int \ln x\,dx=x(\ln x-1)+C_{0}}

generalizes to

∫⋯∫ln⁡xdx⋯dx=xnn!(lnx−∑k=1n1k)+∑k=0n−1Ckxkk!{\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!}}} ,
whereCk{\displaystyle C_{k}} are arbitrary constants of integration.

See also

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References

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