Daftar integral dari fungsi logaritmik

Berikut daftar integral dari fungsi logaritmik. Untuk daftar integral lainnya, lihat tabel integral.

(dengan asumsi ${\displaystyle x>0}$, dan konstanta integrasi tidak diperlihatkankan)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(dengan asumsi ${\displaystyle x>0}$, dan konstanta integrasi tidak diperlihatkankan)

${\displaystyle \int x^{m}\ln x\,dx=x^{m+1}\left({\frac {\ln x}{m+1}-{\frac {1}{(m+1)^{2}\right)}$ untuk ${\displaystyle m\neq -1}$

${\displaystyle \int x^{m}(\ln x)^{n}\,dx={\frac {x^{m+1}(\ln x)^{n}{m+1}-{\frac {n}{m+1}\int x^{m}(\ln x)^{n-1}dx}$ untuk untuk ${\displaystyle m\neq -1}$

${\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}={\frac {(\ln x)^{n+1}{n+1}$ untuk ${\displaystyle n\neq -1}$

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

${\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x^{m}=-{\frac {(\ln x)^{n}{(m-1)x^{m-1}+{\frac {n}{m-1}\int {\frac {(\ln x)^{n-1}dx}{x^{m}$ untuk ${\displaystyle m\neq 1}$

${\displaystyle \int {\frac {x^{m}\,dx}{(\ln x)^{n}=-{\frac {x^{m+1}{(n-1)(\ln x)^{n-1}+{\frac {m+1}{n-1}\int {\frac {x^{m}dx}{(\ln x)^{n-1}$ untuk ${\displaystyle n\neq 1}$

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

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

${\displaystyle \int {\frac {dx}{x\ln \ln x}=\operatorname {li} (\ln x)}$

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

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

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

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

(dengan asumsi ${\displaystyle x>0}$, dan konstanta integrasi tidak diperlihatkankan)

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

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

(dengan asumsi ${\displaystyle x>0}$, dan konstanta integrasi tidak diperlihatkankan)

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

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

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

• (Inggris) 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 dalam buku klasik ini.