Math formulas and cheat sheets for Definite integrals of logarithmic functions
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Definite integrals of logarithmic functions
∫
1
0
x
m
(
ln
x
)
n
d
x
=
(
−
1
)
n
n
!
(
m
+
1
)
n
+
1
,
m
>
−
1
,
n
=
0
,
1
,
2
,
…
∫
1
0
ln
x
1
+
x
d
x
=
−
π
2
12
∫
1
0
ln
x
1
−
x
d
x
=
−
π
2
6
∫
1
0
ln
(
1
+
x
)
x
d
x
=
π
2
12
∫
1
0
ln
(
1
−
x
)
x
d
x
=
−
π
2
6
∫
1
0
ln
x
ln
(
1
+
x
)
d
x
=
2
−
2
ln
2
−
π
2
12
∫
1
0
ln
x
ln
(
1
−
x
)
d
x
=
2
−
π
2
6
∫
∞
0
x
p
−
1
ln
x
1
+
x
d
x
=
−
π
2
csc
(
p
π
)
cot
(
p
π
)
,
0
<
p
<
1
∫
1
0
x
m
−
x
n
ln
x
d
x
=
ln
m
+
1
n
+
1
∫
∞
0
e
−
x
ln
x
d
x
=
−
γ
∫
∞
0
e
−
x
2
ln
x
d
x
=
−
π
√
4
(
γ
+
2
ln
2
)
∫
∞
0
ln
(
e
x
+
1
e
x
−
1
)
d
x
=
π
2
4
∫
π
/
2
0
ln
(
sin
x
)
d
x
=
∫
π
/
2
0
ln
(
cos
x
)
d
x
=
−
π
2
ln
2
∫
π
/
2
0
(
ln
(
sin
x
)
)
2
d
x
=
∫
π
/
2
0
(
ln
(
cos
x
)
)
2
d
x
=
π
2
(
ln
2
)
2
+
π
3
24
∫
π
0
x
ln
(
sin
x
)
d
x
=
−
π
2
2
ln
2
∫
π
/
2
0
sin
x
ln
(
sin
x
)
d
x
=
ln
2
−
1
∫
2
π
0
ln
(
a
+
b
sin
x
)
d
x
=
∫
2
π
0
ln
(
a
+
b
cos
x
)
d
x
=
2
π
ln
(
a
+
a
2
−
b
2
−
−
−
−
−
−
√
)
∫
π
0
ln
(
a
+
b
cos
x
)
d
x
=
π
ln
(
a
+
a
2
−
b
2
−
−
−
−
−
−
√
2
)
∫
π
0
ln
(
a
2
−
2
a
b
cos
x
+
b
2
)
d
x
=
{
2
π
ln
a
2
π
ln
b
a
≥
b
>
0
b
≥
a
>
0
∫
π
/
4
0
ln
(
1
+
tan
x
)
d
x
=
π
8
ln
2
∫
π
2
0
sec
x
ln
(
1
+
b
cos
x
1
+
a
cos
x
)
d
x
=
1
2
(
arccos
2
a
−
arccos
2
b
)