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Dit artikel bevat een lijst van integralen van logaritmische functies . Nota bene dat x niet gedefinieerd is wanneer het kleiner of gelijk is aan nul . Tevens is de integratieconstante C weggelaten.
∫
ln
a
x
d
x
=
x
ln
a
x
−
x
{\displaystyle \int \ln ax\,\mathrm {d} x=x\ln ax-x}
∫
ln
(
a
x
+
b
)
d
x
=
(
a
x
+
b
)
ln
(
a
x
+
b
)
−
(
a
x
)
a
{\displaystyle \int \ln(ax+b)\,\mathrm {d} x={\frac {(ax+b)\ln(ax+b)-(ax)}{a}}}
∫
(
ln
x
)
2
d
x
=
x
(
ln
x
)
2
−
2
x
ln
x
+
2
x
{\displaystyle \int (\ln x)^{2}\,\mathrm {d} x=x(\ln x)^{2}-2x\ln x+2x}
∫
(
ln
x
)
n
d
x
=
x
∑
k
=
0
n
(
−
1
)
n
−
k
n
!
k
!
(
ln
x
)
k
{\displaystyle \int (\ln x)^{n}\,\mathrm {d} x=x\sum _{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}}(\ln x)^{k}}
∫
d
x
ln
x
=
ln
|
ln
x
|
+
ln
x
+
∑
k
=
2
∞
(
ln
x
)
k
k
⋅
k
!
{\displaystyle \int {\frac {\mathrm {d} x}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}
∫
d
x
(
ln
x
)
n
=
−
x
(
n
−
1
)
(
ln
x
)
n
−
1
+
1
n
−
1
∫
d
x
(
ln
x
)
n
−
1
(
voor
n
≠
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {\mathrm {d} x}{(\ln x)^{n-1}}}\qquad ({\mbox{voor }}n\neq 1)}
∫
x
m
ln
x
d
x
=
x
m
+
1
(
ln
x
m
+
1
−
1
(
m
+
1
)
2
)
(
voor
m
≠
−
1
)
{\displaystyle \int x^{m}\ln x\,\mathrm {d} x=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad ({\mbox{voor }}m\neq -1)}
∫
x
m
(
ln
x
)
n
d
x
=
x
m
+
1
(
ln
x
)
n
m
+
1
−
n
m
+
1
∫
x
m
(
ln
x
)
n
−
1
d
x
(
voor
m
≠
−
1
)
{\displaystyle \int x^{m}(\ln x)^{n}\,\mathrm {d} x={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}\,\mathrm {d} x\qquad ({\mbox{voor }}m\neq -1)}
∫
(
ln
x
)
n
d
x
x
=
(
ln
x
)
n
+
1
n
+
1
(
voor
n
≠
−
1
)
{\displaystyle \int {\frac {(\ln x)^{n}\,\mathrm {d} x}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad ({\mbox{voor }}n\neq -1)}
∫
ln
x
n
d
x
x
=
(
ln
x
n
)
2
2
n
(
voor
n
≠
0
)
{\displaystyle \int {\frac {\ln {x^{n}}\,\mathrm {d} x}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}\qquad ({\mbox{voor }}n\neq 0)}
∫
ln
x
d
x
x
m
=
−
ln
x
(
m
−
1
)
x
m
−
1
−
1
(
m
−
1
)
2
x
m
−
1
(
voor
m
≠
1
)
{\displaystyle \int {\frac {\ln x\,\,\mathrm {d} x}{x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}\qquad ({\mbox{voor }}m\neq 1)}
∫
(
ln
x
)
n
d
x
x
m
=
−
(
ln
x
)
n
(
m
−
1
)
x
m
−
1
+
n
m
−
1
∫
(
ln
x
)
n
−
1
d
x
x
m
(
voor
m
≠
1
)
{\displaystyle \int {\frac {(\ln x)^{n}\,\mathrm {d} x}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}\,\mathrm {d} x}{x^{m}}}\qquad ({\mbox{voor }}m\neq 1)}
∫
x
m
d
x
(
ln
x
)
n
=
−
x
m
+
1
(
n
−
1
)
(
ln
x
)
n
−
1
+
m
+
1
n
−
1
∫
x
m
d
x
(
ln
x
)
n
−
1
(
voor
n
≠
1
)
{\displaystyle \int {\frac {x^{m}\,\mathrm {d} x}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}\,\mathrm {d} x}{(\ln x)^{n-1}}}\qquad ({\mbox{voor }}n\neq 1)}
∫
d
x
x
ln
x
=
ln
|
ln
x
|
{\displaystyle \int {\frac {\mathrm {d} x}{x\ln x}}=\ln |\ln x|}
∫
d
x
x
n
ln
x
=
ln
|
ln
x
|
+
∑
k
=
1
∞
(
−
1
)
k
(
n
−
1
)
k
(
ln
x
)
k
k
⋅
k
!
{\displaystyle \int {\frac {\mathrm {d} x}{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!}}}
∫
d
x
x
(
ln
x
)
n
=
−
1
(
n
−
1
)
(
ln
x
)
n
−
1
(
voor
n
≠
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad ({\mbox{voor }}n\neq 1)}
∫
ln
(
x
2
+
a
2
)
d
x
=
x
ln
(
x
2
+
a
2
)
−
2
x
+
2
a
tan
−
1
x
a
{\displaystyle \int \ln(x^{2}+a^{2})\,\mathrm {d} x=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}
∫
x
x
2
+
a
2
ln
(
x
2
+
a
2
)
d
x
=
1
4
ln
2
(
x
2
+
a
2
)
{\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,\mathrm {d} x={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}
∫
ln
(
x
n
)
d
x
=
x
2
(
ln
(
x
n
)
−
n
)
{\displaystyle \int \ln \left({\sqrt {x^{n}}}\right)\,\mathrm {d} x={\frac {x}{2}}(\ln(x^{n})-n)}
∫
sin
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
−
cos
(
ln
x
)
)
{\displaystyle \int \sin(\ln x)\,\mathrm {d} x={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}
∫
cos
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
+
cos
(
ln
x
)
)
{\displaystyle \int \cos(\ln x)\,\mathrm {d} x={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}
∫
e
x
(
x
ln
x
−
x
−
1
x
)
d
x
=
e
x
(
x
ln
x
−
x
−
ln
x
)
{\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\,\mathrm {d} x=e^{x}(x\ln x-x-\ln x)}
∫
1
e
x
(
1
x
−
ln
x
)
d
x
=
ln
x
e
x
{\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\,\mathrm {d} x={\frac {\ln x}{e^{x}}}}
∫
(
ln
x
)
x
d
x
=
(
ln
x
)
x
−
1
+
(
ln
(
ln
x
)
)
(
ln
x
)
x
{\displaystyle \int (\ln x)^{x}\,\mathrm {d} x=(\ln x)^{x-1}+(\ln(\ln x))(\ln x)^{x}}
∫
e
x
(
1
ln
x
−
1
x
ln
2
x
)
d
x
=
e
x
ln
x
{\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x\ln ^{2}x}}\right)\,\mathrm {d} x={\frac {e^{x}}{\ln x}}}
Referenties Milton Abramowitz & Irene A. Stegun - Handbook of Mathematical Functions with Formulas , Graphs, and Mathematical Tables, 1964
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