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Lesson4bIntegralsoflnx.docxIntegrals of ln NAME: _____________________________________________________________
Find the integral of the following natural logarithmic functions by SUBSTITUTION METHOD.
RECOGNIZE THE QUOTIENT FORM OF the LN rule as of following:
# 1 to 12 let U = the denominator for SUBSTITUTION
Use Long division before integrating for # 13 and 14 below before integrating
Use U-sub for ln(expression) rule
(
)
3.
5
:ln5
x
x
x
e
dx
e
ansec
-
-+
ò
2
2
2
4.
5
:ln5
x
x
x
e
dx
e
ansec
-
-+
ò
(
)
4
5
5
1
5
5.
7
:ln7
x
dx
x
ansxc
+
++
ò
(
)
sin
6.
cos
:lncos
x
dx
x
ansxc
-+
ò
(
)
sin
sin
sin
cos
7.
2
:ln2
x
x
x
xe
dx
e
anseC
+
++
ò
(
)
1
8.
(ln5)
:lnln5
dx
xx
ansxc
×+
++
ò
(
)
1
5
cos(5)
9.
sin(5)5
:lnsin(5)5
x
dx
x
ansxc
+
++
ò
(
)
21
33
1
3
1
10.
:3ln1
dx
xx
ansxc
æö
+
ç÷
èø
++
ò
(
)
()()()
b
b
x
a
a
fdxFxFbFa
==-
ò
2
1
8.
ln
:ln2
e
e
dx
xx
ans
ò
(
)
4
2
3
2
4
3
csc
12.
3cot
:ln4ln3ln
x
dx
x
ans
p
p
-
-=
ò
2
2
32
1
6
2
13.
:4ln(1)
xx
x
x
dx
ansxxc
-+
+
-+++
ò
(
)
2
2
2
1
2
1
14.
1
:ln1
xx
dx
x
ansxxc
++
+
+++
ò
(
)
2
2
15.
56
:ln6
xx
xx
x
ee
dx
ee
Factoraspolynomial
ansec
+
--
-+
ò
(
)
2
2
(tan3)sec
16.
tan2tan3
:lntan1
xx
dx
xx
Factoraspolynomial
ansxc
-
--
++
ò
(
)
(
)
'
1
'&ln
x
x
U
dUUdxdxdUUC
UU
===+
òò
cot
17.
ln(sin)
:ln(ln(sin))
x
dx
x
ansxc
+
ò
tan
18.
lnsec
:ln(ln(sec))
x
dx
x
ansxc
+
ò
5
5
1
5
1
19.
ln
:ln(ln))
dx
xx
ansxc
+
ò
(
)
2
1
2
20.ln(sin)cot
:ln(sin)
xxdx
ansxc
+
ò
(
)
2
1
2
21.ln(cos)tan
:ln(cos)
xxdx
ansc
-+
ò
(
)
3
2
3
1
6
ln
22.
:ln
x
dx
x
ansxc
+
ò
(
)
(
)
(
)
2
1
6
ln35
23.
(35)
:ln35
x
dx
x
ansxc
+
+
++
ò
(
)
(
)
(
)
(
)
2
1
2
ln1
24.
1
:ln1
xx
x
x
ee
dx
e
ansec
+
+
++
ò
2
3
3
1
3
1.
3
:ln(3)
x
dx
x
ansxc
-
--+
ò
2
csc
2.
cot
:ln|cot|
t
dt
t
ansc
-+
ò
