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17.6 Substitution in Integrals
Let
be a nice function on an interval
. Then if
is any
antiderivative
for
, we have
by the fundamental theorem of calculus. We saw in (17.32) that under
suitable hypotheses on
,
is an antiderivative for
.
Hence
Hence we can find
by the following
ritual:
Let
. When
then
and when
then
. Also
. Hence
17.51
Example.
To find

.
Let
. When
, then
, and when
, then
.
Also
, so
We saw in (17.36) that if
is an inverse function for
, then an
antiderivative for
is
, where
is an antiderivative for
. Thus
The ritual
for finding
in this case is:
Let
. Then
and
. When
then
, and
when
then
. Thus
where
is an antiderivative for
.
17.52
Example.
To find

.
Let
. When
then
, and when
then
.
Also
, so
.
17.53
Exercise.
A
Find the following integrals:
- a)
-
.
- b)
-
.
- c)
-
.
17.54
Exercise.
A
Find the area of the shaded region, bounded by the ellipse

and the lines

.
17.55
Example.
In practice I would find many of the antiderivatives and integrals
discussed
in this chapter by computer. For example, using Maple, I would find
> int(sqrt(a^2+x^2),x);
> int(sin(sqrt(x)),x=0..Pi^2);
> int(sqrt(4 - x^2),x=-1..1);
> int( (sec(x))^3,x);
> int(exp(a*x)*cos(b*x),x);
Next: 17.7 Rational Functions
Up: 17. Antidifferentiation Techniques
Previous: 17.5 Trigonometric Substitution
  Index
Ray Mayer
2007-09-07