Usage If it is impossible to find the indefinite integral of , then the following might help.
NOTE Let
. Then
. Now the limit of intervals must be changed from
to
and
to
.
.
Integration by Parts
Let
be differentiable on the closed interval
. Then
Usage We let
be
.
SOLUTION 1. Let
. Then
. Thus we can express the integrand as
. Furthermore, the limit of integration becomes
.
Thus,
2.
Then
If we let , then
and
. Thus we can not solve this by u-substitution directly.
Let
.
SOLUTION Let
. Then,
and
.
. Now need to express
by
.
,
. Thus,
Since
,
. Thus,
. Note
. Thus
.
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Since the limit of integral is
,
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u-substitution |
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In u-substitution, complexity of integration depends on the choice of ![]() ![]() |
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Properties of Definite Integral Suppose that is continuous on the limit of integration.
1. If is even function, then
2. If is odd function, then
3.
where
Even/Odd Functions is even function means that
and the graph of a function
is symmetric with respect to the
-axis. Thus,
.
is odd function means that
and the graph of a function
is symmetric with respect to the
-axis. Thus,
.
Since
, we have
and
.
,
,
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4. By 3.
. We show
.
For ,
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Exercise A
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(a) Show that is an odd function
(b) Show that is even function implies that
is an odd function.
(c) Show that
implies that
(d) Show that can be represented by a sum and a difference of functions