7.1 Integration by Parts
Review the integrals shown on page 469 and integration by substitution.
Examples:
1
( )
( )
(sin
)(cos
)
x
x
e
e
a
dx
b
x
x
+
∫
∫
dx
Remember - integration is done by various techniques and is not as
straightforward as differentiation.
Typically, selecting the proper technique is the most difficult aspect of
integration.
Large numbers of practice problems is the best solution to the difficulty you may
encounter.
The integration rule that corresponds to the product rule for differentiation is
integration by parts.
INTEGRATION BY PARTS.
u dv
uv
v du
=
−
∫
∫
OBJECT:
select a substitution that yields a
simpler
integral.
GENERALLY:
select
u = f(x)
to be a function that becomes simpler when
differentiated (or at least not more complicated).
Examples (Note Example 2 on page 471)
( )
cos
( )
x
a
x
x d
b
xe dx

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