قائمة تكاملات الدوال اللوغاريتمية

دوال لوغاريتيمة

${\displaystyle{\displaystyle\int\ln cx\;dx=x\ln cx-x}}$

${\displaystyle{\displaystyle\int\ln(ax+b)\;dx={\frac{(ax+b)\ln(ax+b)-(ax+b)}{a}}}}$

${\displaystyle{\displaystyle\int(\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x}}$

${\displaystyle{\displaystyle\int(\ln cx)^{n}\;dx=x(\ln cx)^{n}-n\int(\ln cx)^{n-1}dx}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\ln x}}=\ln|\ln x|+\ln x+\sum_{k=2}^{\infty}{\frac{(\ln x)^{k}}{k\cdot k!}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{(\ln x)^{n}}}=-{\frac{x}{(n-1)(\ln x)^{n-1}}}+{\frac{1}{n-1}}\int{\frac{dx}{(\ln x)^{n-1}}}\qquad{\mbox{(for }}n\neq 1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int x^{m}\ln x\;dx=x^{m+1}\left({\frac{\ln x}{m+1}}-{\frac{1}{(m+1)^{2}}}\right)\qquad{\mbox{(for }}m\neq-1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int x^{m}(\ln x)^{n}\;dx={\frac{x^{m+1}(\ln x)^{n}}{m+1}}-{\frac{n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\qquad{\mbox{(for }}m\neq-1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{(\ln x)^{n}\;dx}{x}}={\frac{(\ln x)^{n+1}}{n+1}}\qquad{\mbox{(for }}n\neq-1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{\ln{x^{n}}\;dx}{x}}={\frac{(\ln{x^{n}})^{2}}{2n}}\qquad{\mbox{(for }}n\neq 0{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{\ln x\,dx}{x^{m}}}=-{\frac{\ln x}{(m-1)x^{m-1}}}-{\frac{1}{(m-1)^{2}x^{m-1}}}\qquad{\mbox{(for }}m\neq 1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{(\ln x)^{n}\;dx}{x^{m}}}=-{\frac{(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac{n}{m-1}}\int{\frac{(\ln x)^{n-1}dx}{x^{m}}}\qquad{\mbox{(for }}m\neq 1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{x^{m}\;dx}{(\ln x)^{n}}}=-{\frac{x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac{m+1}{n-1}}\int{\frac{x^{m}dx}{(\ln x)^{n-1}}}\qquad{\mbox{(for }}n\neq 1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x\ln x}}=\ln\left|\ln x\right|}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x^{n}\ln x}}=\ln\left|\ln x\right|+\sum_{k=1}^{\infty}(-1)^{k}{\frac{(n-1)^{k}(\ln x)^{k}}{k\cdot k!}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x(\ln x)^{n}}}=-{\frac{1}{(n-1)(\ln x)^{n-1}}}\qquad{\mbox{(for }}n\neq 1{\mbox{)}}}}$

${\displaystyle{\displaystyle\int\ln(x^{2}+a^{2})\;dx=x\ln(x^{2}+a^{2})-2x+2a\tan^{-1}{\frac{x}{a}}}}$

${\displaystyle{\displaystyle\int{\frac{x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\;dx={\frac{1}{4}}\ln^{2}(x^{2}+a^{2})}}$

${\displaystyle{\displaystyle\int\sin(\ln x)\;dx={\frac{x}{2}}(\sin(\ln x)-\cos(\ln x))}}$

${\displaystyle{\displaystyle\int\cos(\ln x)\;dx={\frac{x}{2}}(\sin(\ln x)+\cos(\ln x))}}$

${\displaystyle{\displaystyle\int e^{x}\left(x\ln x-x-{\frac{1}{x}}\right)\;dx=e^{x}(x\ln x-x-\ln x)}}$

${\displaystyle{\displaystyle\int{\frac{1}{e^{x}}}\left({\frac{1}{x}}-\ln x\right)\;dx={\frac{\ln x}{e^{x}}}}}$

${\displaystyle{\displaystyle\int e^{x}\left({\frac{1}{\ln x}}-{\frac{1}{x\ln^{2}x}}\right)\;dx={\frac{e^{x}}{\ln x}}}}$

 المصادر

• 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 in this classic book.