Integration by Parts
Integration by Parts is the name of a technique for integrating certain types if
functions. It is based on the Product Rule for differentiation, namely
(
29
uv
uv
u v
′
′
′
=
+
.
Integrating this expression (assuming the appropriate antiderivatives exist) yields
uv
uv dx
u vdx
′
′
=
+
∫
∫
udv
vdu
=
+
∫
∫
.
Writing this last expression in a different order yields the formula
udv
uv
vdu
=

∫
∫
.
In applying this formula keep the following things in mind:
One must identify the choices of
u
and
v
. Some choices may work and
others may not.
For the definite integral, the formula is
(
29
b
b
b
a
a
a
udv
uv
vdu
=

∫
∫
.
Choosing
u
and
v
is a guess. You get better at making successful guesses
by gaining experience through working problems.
The examples below illustrate these ideas.
Example 1
: Evaluate
ln
x
xdx
∫
.
Solution
: Choose
ln
,
u
x dv
xdx
=
=
. Then
2
1
,
2
dx
du
v
x
x
=
=
.
(
We usually write this as
2
ln
,
1
,
2
u
x dv
xdx
dx
du
v
x
x
=
=
=
=
. Notice that
ln
x
xdx
udv
=
.) Then
ln
x
xdx
udv
uv
vdu
=
=

∫
∫
∫
2
2
1
1
ln
2
2
dx
x
x
x
x
=

∫
2
1
1
ln
2
2
x
x
xdx
=

∫
2
2
1
1
1
ln
2
2
2
x
x
x
C
=

+
(
29
2
1
2ln
1
4
x
x
C
=

+
.
Example 2
: Evaluate
2
sec
t
tdt
∫
Solution
: Let
2
,
sec
,
tan
u
t dv
tdt
du
dt v
t
=
=
=
=
.
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 Spring '11
 BUEKMAN
 Fundamental Theorem Of Calculus, Integration By Parts, Product Rule, Sin, dx

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