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.
∫
log
a
x
d
x
=
x
ln
x
−
x
ln
a
∫
ln
a
x
d
x
=
x
ln
a
x
−
x
∫
ln
(
a
x
+
b
)
d
x
=
(
a
x
+
b
)
ln
(
a
x
+
b
)
−
a
x
a
∫
(
ln
x
)
2
d
x
=
x
(
ln
x
)
2
−
2
x
ln
x
+
2
x
∫
(
ln
x
)
n
d
x
=
x
∑
k
=
0
n
(
−
1
)
n
−
k
n
!
k
!
(
ln
x
)
k
=
Θ
(
x
(
ln
x
)
n
)
∫
d
x
ln
x
=
ln
|
ln
x
|
+
ln
x
+
∑
k
=
2
∞
(
ln
x
)
k
k
⋅
k
!
∫
d
x
ln
x
=
li
(
x
)
= the logarithmic integral (asymptotically,
li
(
x
)
=
Θ
(
x
ln
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
)
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
)
∫
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
)
∫
(
ln
x
)
n
d
x
x
=
(
ln
x
)
n
+
1
n
+
1
(for
n
≠
−
1
)
∫
ln
x
n
d
x
x
=
(
ln
x
n
)
2
2
n
(for
n
≠
0
)
∫
ln
x
d
x
x
m
=
−
ln
x
(
m
−
1
)
x
m
−
1
−
1
(
m
−
1
)
2
x
m
−
1
(for
m
≠
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
(for
m
≠
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
(for
n
≠
1
)
∫
d
x
x
ln
x
=
ln
|
ln
x
|
∫
d
x
x
ln
x
ln
ln
x
=
ln
|
ln
|
ln
x
|
|
, etc.
∫
d
x
x
ln
ln
x
=
li
(
ln
x
)
where li is the logarithmic integral.
∫
d
x
x
n
ln
x
=
ln
|
ln
x
|
+
∑
k
=
1
∞
(
−
1
)
k
(
n
−
1
)
k
(
ln
x
)
k
k
⋅
k
!
∫
d
x
x
(
ln
x
)
n
=
−
1
(
n
−
1
)
(
ln
x
)
n
−
1
(for
n
≠
1
)
∫
ln
(
x
2
+
a
2
)
d
x
=
x
ln
(
x
2
+
a
2
)
−
2
x
+
2
a
tan
−
1
x
a
∫
x
x
2
+
a
2
ln
(
x
2
+
a
2
)
d
x
=
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
)
)
∫
cos
(
ln
x
)
d
x
=
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
)
∫
1
e
x
(
1
x
−
ln
x
)
d
x
=
ln
x
e
x
∫
e
x
(
1
ln
x
−
1
x
ln
2
x
)
d
x
=
e
x
ln
x
For
n
consecutive integrations, the formula
∫
ln
x
d
x
=
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
!
