Section 5.5: The Substitution Rule
Instructor: Ms. Hoa Nguyen
([email protected])
The Substitution Rule
The idea behind the Substitution Rule is to replace a relatively complicated integral with a
simpler integral by changing from one variable to another.
Suppose we want to evaluate
f
(
x
)
dx
which cannot be evaluated from the formulas in the
previous sections. The Substitution Rule suggests to change from the variable
x
to a new
variable
u
in the following steps:
•
Let
u
=
g
(
x
) (
g
(
x
) is a differentiable function whose range is a domain of
f
).
Try
choosing
g
(
x
) to be some part of the integrand
f
(
x
).
•
Then the differential of
u
is
du
=
g
(
x
)
dx
.
•
Substitute
f
(
x
)
dx
by
u
and
du
, if possible.
In other words,
f
(
g
(
x
))
g
(
x
)
dx
=
f
(
u
)
du
. Otherwise, try a new substitution.
•
After evaluating the new integral as a function of
u
, we need to replace
u
by
g
(
x
) to
return to the original variable
x
.
Remark
: The main challenge in using the rule is to think of an appropriate substitution.

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