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The following is a list of integrals (antiderivative functions) of logarithmic functions . For a complete list of integral functions, see list of integrals .
Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
Integrals involving only logarithmic functions
∫
log
a
x
d
x
=
x
log
a
x
−
x
ln
a
=
x
ln
x
−
x
ln
a
{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}={\frac {x\ln x-x}{\ln a}}}
∫
ln
(
a
x
)
d
x
=
x
ln
(
a
x
)
−
x
{\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x}
∫
ln
(
a
x
+
b
)
d
x
=
(
a
x
+
b
)
ln
(
a
x
+
b
)
−
(
a
x
+
b
)
a
{\displaystyle \int \ln(ax+b)\,dx={\frac {(ax+b)\ln(ax+b)-(ax+b)}{a}}}
∫
(
ln
x
)
2
d
x
=
x
(
ln
x
)
2
−
2
x
ln
x
+
2
x
{\displaystyle \int (\ln x)^{2}\,dx=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}\,dx=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 {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}
∫
d
x
ln
x
=
li
(
x
)
{\displaystyle \int {\frac {dx}{\ln x}}=\operatorname {li} (x)}
logarithmic integral .
∫
d
x
(
ln
x
)
n
=
−
x
(
n
−
1
)
(
ln
x
)
n
−
1
+
1
n
−
1
∫
d
x
(
ln
x
)
n
−
1
(for
n
≠
1
)
{\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{)}}}
∫
ln
f
(
x
)
d
x
=
x
ln
f
(
x
)
−
∫
x
f
′
(
x
)
f
(
x
)
d
x
(for differentiable
f
(
x
)
>
0
)
{\displaystyle \int \ln f(x)\,dx=x\ln f(x)-\int x{\frac {f'(x)}{f(x)}}\,dx\qquad {\mbox{(for differentiable }}f(x)>0{\mbox{)}}}
Integrals involving logarithmic and power functions
∫
x
m
ln
x
d
x
=
x
m
+
1
(
ln
x
m
+
1
−
1
(
m
+
1
)
2
)
(for
m
≠
−
1
)
{\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{)}}}
∫
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
(for
m
≠
−
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\qquad {\mbox{(for }}m\neq -1{\mbox{)}}}
∫
(
ln
x
)
n
d
x
x
=
(
ln
x
)
n
+
1
n
+
1
(for
n
≠
−
1
)
{\displaystyle \int {\frac {(\ln x)^{n}\,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
−
1
(
m
−
1
)
2
x
m
−
1
(for
m
≠
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}}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}
∫
(
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
(for
m
≠
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}}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}
∫
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
(for
n
≠
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}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
d
x
x
ln
x
=
ln
|
ln
x
|
{\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}
∫
d
x
x
ln
x
ln
ln
x
=
ln
|
ln
|
ln
x
|
|
{\displaystyle \int {\frac {dx}{x\ln x\ln \ln x}}=\ln \left|\ln \left|\ln x\right|\right|}
∫
d
x
x
ln
ln
x
=
li
(
ln
x
)
{\displaystyle \int {\frac {dx}{x\ln \ln x}}=\operatorname {li} (\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 {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!}}}
∫
d
x
x
(
ln
x
)
n
=
−
1
(
n
−
1
)
(
ln
x
)
n
−
1
(for
n
≠
1
)
{\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
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})\,dx=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})\,dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}
Integrals involving logarithmic and trigonometric functions
∫
sin
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
−
cos
(
ln
x
)
)
{\displaystyle \int \sin(\ln x)\,dx={\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)\,dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}
Integrals involving logarithmic and exponential functions
∫
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)\,dx=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)\,dx={\frac {\ln x}{e^{x}}}}
∫
e
x
(
1
ln
x
−
1
x
(
ln
x
)
2
)
d
x
=
e
x
ln
x
{\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x(\ln x)^{2}}}\right)\,dx={\frac {e^{x}}{\ln x}}}
n consecutive integrationsFor
n
{\displaystyle n}
∫
ln
x
d
x
=
x
(
ln
x
−
1
)
+
C
0
{\displaystyle \int \ln x\,dx=x(\ln x-1)+C_{0}}
generalizes to
∫
⋯
∫
ln
x
d
x
⋯
d
x
=
x
n
n
!
(
ln
x
−
∑
k
=
1
n
1
k
)
+
∑
k
=
0
n
−
1
C
k
x
k
k
!
{\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!}}}
See also References
