The following is a list of integrals (antiderivative functions) of logarithmic functions.
Berikut daftar integral dari fungsi logaritmik.
For a complete list of integral functions, see list of integrals.
Untuk daftar integral lainnya, lihat tabel integral.
∫ log a ⁡ x d xx log a ⁡ x − x ln ⁡ ax ln ⁡ a ( ln ⁡ x − 1 ) x\,dx=x\log _{a}x-{\frac {x}{\ln a}}={\frac {x}{\ln a}}(\ln x-1)} ∫ ln ⁡ ( a x )
(dengan asumsi x > 0 , dan konstanta integrasi tidak diperlihatkankan) ∫ ln ⁡ c x d xx ln ⁡ c x − x ∫ ln ⁡ ( a x + b )
d xa x + b a ( ln ⁡ ( a x + b ) − 1 ) {a}}(\ln(ax+b)-1)} ∫ ( ln ⁡ x ) 2 d xx ( ln ⁡ x ) 2 − 2 x ln ⁡ x + 2 x \,dx=x(\ln x)^{2}-2x\ln x+2x} ∫ ln n ⁡ x d x( − 1 ) n x ∑ k0 n n !
d xx ln ⁡ ( a x + b ) − x + b a ln ⁡ ( a x + b ) {a}}\ln(ax+b)} ∫ ( ln ⁡ x ) 2 d xx ( ln ⁡ x ) 2 − 2 x ln ⁡ x + 2 x \;dx=x(\ln x)^{2}-2x\ln x+2x} ∫ ( ln ⁡ c x ) n d xx ( ln ⁡ c x ) n − n ∫ ( ln ⁡ c x ) n − 1 d x \;dx=x(\ln cx)^{n}-n\int (\ln cx)^{n-1}dx} ∫ d x ln ⁡ xln ⁡ | ln ⁡ x | + ln ⁡ x + ∑ i2 ∞ ( ln ⁡ x )
k ! ( − ln ⁡ x ) k {x}\,dx}=(-1)^{n}\,x\,\sum _{k=0}^{n}{\frac {n!}{k!}}(-\ln x)^{k}} ∫ d x ln ⁡ xln ⁡ | ln ⁡ x | + ln ⁡ x + ∑ k2 ∞ ( ln ⁡ x ) k k ⋅ k !
i i ⋅ i ! {\ln x}}=\ln |\ln x|+\ln x+\sum _{i=2}^{\infty }{\frac {(\ln x)^{i}}{i\cdot i!}}} ∫ d x ( ln ⁡ x ) n− x ( n − 1 ) ( ln ⁡ x ) n − 1 + 1 n − 1 ∫ d x ( ln ⁡ x ) n − 1 {(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}} untuk n ≠ 1 ∫ x m ln ⁡ x d
{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}} ∫ d x ln ⁡ xli ⁡ ( x ) {\ln x}}=\operatorname {li} (x)} , the logarithmic integral.
d xln ⁡ x e x {e^{x}}}\left({\frac {1}{x}}-\ln x\right)\;dx={\frac {\ln x}{e^{x}}}} ∫ e x ( 1 ln ⁡ x − 1 x ln 2 ⁡ x )
∫ d x ( ln ⁡ x ) n− x ( n − 1 ) ( ln ⁡ x ) n − 1 + 1 n − 1 ∫ d x ( ln ⁡ x ) n − 1 (for n ≠ 1 ) {(\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 )
xx m + 1 ( ln ⁡ x ) n m + 1 − n m + 1 ∫ x m ( ln ⁡ x ) n − 1 d x (\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 m ≠ − 1 ∫ ( ln ⁡ x ) n d x x( ln ⁡ x ) n + 1 n + 1 \;dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}} untuk n ≠ − 1 ∫ ln ⁡ x n d x x( ln ⁡ x n ) 2 2 n }\;dx}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}} untuk n ≠ 0 ∫ ln ⁡ x d x x m− ln ⁡ x ( m − 1 )
(for m ≠ − 1 ) \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 xx m + 1 ( ln ⁡ x ) n m + 1 − n m + 1 ∫ x m ( ln ⁡ x ) n − 1 d x (for m ≠ − 1 ) (\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 ) \,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}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)} untuk m ≠ − 1 ∫ x m ( ln ⁡ x ) n d xx m + 1 ( ln ⁡ x ) n m + 1 − n m + 1 ∫ x m ( ln ⁡ x ) n − 1 d x (\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 m ≠ − 1 ∫ ( ln ⁡ x ) n d x x( ln ⁡ x ) n + 1 n + 1 \,dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}} untuk n ≠ − 1 ∫ ln ⁡ x d x x m− ln ⁡ x ( m − 1 )
x m − 1 − 1 ( m − 1 ) 2 x m − 1 (for m ≠ 1 ) {x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {
x m − 1 − 1 ( m − 1 ) 2 x m − 1 {x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}} untuk m ≠ 1 ∫ ( ln ⁡ x ) n d x x m− ( ln ⁡ x ) n ( m − 1 )
d x1 4 ln 2 ⁡ ( x 2 + a 2 ) {x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})} ∫ sin ⁡ ( ln ⁡ x )
d x1 4 ln 2 ⁡ ( x 2 + a 2 ) {x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})} (dengan asumsi x > 0 , dan konstanta integrasi tidak diperlihatkankan) ∫ sin ⁡ ( ln ⁡ x )
d xx 2 ( sin ⁡ ( ln ⁡ x ) + cos ⁡ ( ln ⁡ x ) ) {2}}(\sin(\ln x)+\cos(\ln x))}
d xx 2 ( sin ⁡ ( ln ⁡ x ) − cos ⁡ ( ln ⁡ x ) ) {2}}(\sin(\ln x)-\cos(\ln x))} ∫ cos ⁡ ( ln ⁡ x )
d xe x ( x ln ⁡ x − x − ln ⁡ 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 xln ⁡ x e x {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 xe x ln ⁡ x \left({\frac {1}{\ln x}}-{\frac {1}{x(\ln x)^{2}}}\right)\,dx={\frac {e^{x}}{\ln x}}} For n consecutive integrations, the formula ∫ ln ⁡ x d xx ( ln ⁡ x − 1 ) + C 0 } generalizes to ∫ ⋯ ∫ ln ⁡ x d x ⋯ d xx n n !
d xln ⁡ x e x {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 )
( ln x − ∑ k1 n 1 k ) + ∑ k0 n − 1 C k x k k ! }{n!}}\left(\ln \,x-\sum _{k=1}^{n}{\frac {1}{k}}\right)+\sum _{k=0}^{n-1}C_{k}{\frac {x^{k}}{k!}}} , where C k } are arbitrary constants of integration.
i i ⋅ i ! {\ln x}}=\ln |\ln x|+\ln x+\sum _{i=2}^{\infty }{\frac {(\ln x)^{i}}{i\cdot i!}}} ∫ d x ( ln ⁡ x ) n− x ( n − 1 ) ( ln ⁡ x ) n − 1 + 1 n − 1 ∫ d x ( ln ⁡ x ) n − 1 {(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}} untuk n ≠ 1 ∫ x m ln ⁡ x d
A few integrals are listed on page 69.
A few integrals are listed on page 69 dalam buku klasik ini.