More Substitution Rule
In order to allow these pages to be displayed on the web
we�ve broken the substitution rule examples into two sections. The previous
section contains the introduction to the substitution rule and some fairly basic
examples. The examples in this section tend towards the slightly more difficult
side. Also, we�ll not be putting quite as much explanation into the solutions
here as we did in the previous section.
In the first couple of sets of problems in this section the
difficulty is not with the actual integration itself, but with the set up for
the integration. Most of the integrals are fairly simple and most of the
substitutions are fairly simple. The problems arise in getting the integral set
up properly for the substitution(s) to be done. Once you see how these are done
it�s easy to see what you have to do, but the first time through these can cause
problems if you aren�t on the lookout for potential problems.
Example Evaluate
each of the following integrals.
(a)
(b)
(c)
Solution
(a)
This first integral has two terms in it and both will
require the same substitution. This means that we won�t have to do anything
special to the integral. One of the more common �mistakes� here is to break the
integral up and do a separate substitution on each part. This isn�t really
mistake but will definitely increase the amount of work we�ll need to do. So,
since both terms in the integral use the same substitution we�ll just do
everything as a single integral using the following substitution.
The integral is then,
Often a substitution can be used multiple times in an
integral so don�t get excited about that if it happens. Also note that since
there was a
 in
front of the whole integral there must also be a
 in
front of the answer from the integral.
(b)
This integral is similar to the previous one, but it might
not look like it at first glance. Here is the substitution for this problem,
We�ll plug the substitution into the problem twice (since
there are two cosines) and will only work because there is a sine multiplying
everything. Without that sine in front we would not be able to use this
substitution.
The integral in this case is,
Again, be careful with the minus sign in front of the whole
integral.
(c)
It should be fairly clear that each term in this integral
will use the same substitution, but let�s rewrite things a little to make things
really clear.
Since each term had an x in it and we�ll need that
for the differential we factored that out of both terms to get it into the
front. This integral is now very similar to the previous one. Here�s the
substitution.
The integral is,
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