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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