नीचे लघुगणकीय फलनों के समाकलों की सूची दी गयी है।

*ध्यान दें : इस पूरे लेख में यह माना गया है कि*  *x* > 0। इसके अलावा समाकलन स्थिरांक को लिखने के बजाय छोड दिया गया है।

केवल लघुगणकीय फलन वाले समाकल
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{\displaystyle \int \log \_{a}x\,dx={\frac {x\ln x-x}{\ln a}}}
{\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x}
{\displaystyle \int \ln(ax+b)\,dx={\frac {(ax+b)\ln(ax+b)-ax}{a}}}
{\displaystyle \int (\ln x)^{2}\,dx=x(\ln x)^{2}-2x\ln x+2x}
{\displaystyle \int (\ln x)^{n}\,dx=x\sum \_{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}}(\ln x)^{k}}
{\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum \_{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}
{\displaystyle \int {\frac {dx}{\ln x}}=\operatorname {li} (x)}, the logarithmic integral.
{\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{)}}}

लघुगणक तथा घात वाले फलनों के समाकलन
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{\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{)}}}
{\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{)}}}
{\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
{\displaystyle \int {\frac {\ln {x^{n}}\,dx}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}\qquad {\mbox{(for }}n\neq 0{\mbox{)}}}
{\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{)}}}
{\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{)}}}
{\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{)}}}
{\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}
{\displaystyle \int {\frac {dx}{x\ln x\ln \ln x}}=\ln \left|\ln \left|\ln x\right|\right|}, etc.
{\displaystyle \int {\frac {dx}{x\ln \ln x}}=\operatorname {li} (\ln x)}
{\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!}}}
{\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
{\displaystyle \int \ln(x^{2}+a^{2})\,dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}
{\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}

लघुगणकीय तथा त्रिकोणमितीय फलनों से युक्त फलनों के समाकलन
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{\displaystyle \int \sin(\ln x)\,dx={\frac {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
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{\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\,dx=e^{x}(x\ln x-x-\ln x)}
{\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\,dx={\frac {\ln x}{e^{x}}}}
{\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x(\ln x)^{2}}}\right)\,dx={\frac {e^{x}}{\ln x}}}

*n* क्रमागत समाकल
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{\displaystyle n} क्रमागत समाकल (consecutive integrations) के लिए निम्नलिखित सूत्र का प्रयोग करने पर

{\displaystyle \int \ln x\,dx=x(\ln x-1)+C\_{0}}
निम्नलिखित सामान्यीकरण प्राप्त होता है-

{\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!}}}

सन्दर्भ
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* 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.

इन्हें भी देखें
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* समाकल सूची

| * दे
* वा
* सं
समाकल सूची |
| --- |
| 
* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

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