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

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 .

Let . When then , and when then . Also , so .

`> 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);`