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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