We now need to go back and revisit the substitution rule as
it applies to definite integrals. At
some level there really isn’t a lot to do in this section. Recall that the first step in doing a
definite integral is to compute the indefinite integral and that hasn’t
changed. We will still compute the
indefinite integral first. This means
that we already know how to do these. We
use the substitution rule to find the indefinite integral and then do the
evaluation.
There are however, two ways to deal with the evaluation
step. One of the ways of doing the
evaluation is the probably the most obvious at this point, but also has a point
in the process where we can get in trouble if we aren’t paying attention.
Let’s work an example illustrating both ways of doing the
evaluation step.
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Example 1 Evaluate
the following definite integral.
Solution
Let’s start off looking at the first way of dealing with
the evaluation step. We’ll need to be
careful with this method as there is a point in the process where if we
aren’t paying attention we’ll get the wrong answer.
Solution 1 :
We’ll first need to compute the indefinite integral using
the substitution rule. Note however,
that we will constantly remind ourselves that this is a definite integral by
putting the limits on the integral at each step. Without the limits it’s easy to forget that
we had a definite integral when we’ve gotten the indefinite integral
computed.
In this case the substitution is,
Plugging this into the integral gives,
Notice that we didn’t do the evaluation yet. This is where the potential problem arises
with this solution method. The limits
given here are from the original integral and hence are values of t.
We have u’s in our
solution. We can’t plug values of t in for u.
Therefore, we will have to go back to t’s before we do the substitution. This is the standard step in the
substitution process, but it is often forgotten when doing definite
integrals. Note as well that in this
case, if we don’t go back to t’s we
will have a small problem in that one of the evaluations will end up giving
us a complex number.
So, finishing this problem gives,
So, that was the first solution method. Let’s take a look at the second method.
Solution 2 :
Note that this solution method isn’t really all that
different from the first method. In
this method we are going to remember that when doing a substitution we want
to eliminate all the t’s in the
integral and write everything in terms of u.
When we say all here we really mean all. In other words, remember that the limits on
the integral are also values of t
and we’re going to convert the limits into u values. Converting the
limits is pretty simple since our substitution will tell us how to relate t and u so all we need to do is plug in the original t limits into the substitution and
we’ll get the new u limits.
Here is the substitution (it’s the same as the first
method) as well as the limit conversions.
The integral is now,
As with the first method let’s pause here a moment to
remind us what we’re doing. In this
case, we’ve converted the limits to u’s
and we’ve also got our integral in terms of u’s and so here we can just plug the limits directly into our
integral. Note that in this case we
won’t plug our substitution back in.
Doing this here would cause problems as we would have t’s in the integral and our limits
would be u’s. Here’s the rest of this problem.
We got exactly the same answer and this time didn’t have
to worry about going back to t’s in
our answer.
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So, we’ve seen two solution techniques for computing
definite integrals that require the substitution rule. Both are valid solution methods and each have
their uses. We will be using the second
exclusively however since it makes the evaluation step a little easier.
Let’s work some more examples.
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Example 2 Evaluate
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
Since we’ve done quite a few substitution rule integrals
to this time we aren’t going to put a lot of effort into explaining the
substitution part of things here.
(a)
The substitution and converted limits are,
Sometimes a limit will remain the same after the
substitution. Don’t get excited when
it happens and don’t expect it to happen all the time.
Here is the integral,
Don’t get excited about large numbers for answers
here. Sometime they are. That’s life.
[Return to Problems]
(b)
Here is the substitution and converted limits for this
problem,
The integral is then,
[Return to Problems]
(c)
This integral needs to be split into two integrals since
the first term doesn’t require a substitution and the second does.
Here is the substitution and converted limits for the
second term.
Here is the integral.
[Return to Problems]
(d)
This integral will require two substitutions. So first split up the integral so we can do
a substitution on each term.
There are the two substitutions for these integrals.
Here is the integral for this problem.
[Return to Problems]
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The next set of examples is designed to make sure that we
don’t forget about a very important point about definite integrals.
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Example 3 Evaluate
each of the following.
(a) [Solution]
(b) [Solution]
Solution
(a)
Be careful with this integral. The denominator is zero at and both of these are in the interval of
integration. Therefore, this integrand
is not continuous in the interval and so the integral can’t be done.
Be careful with definite integrals and be on the lookout
for division by zero problems. In the
previous section they were easy to spot since all the division by zero
problems that we had there were at zero.
Once we move into substitution problems however they will not always
be so easy to spot so make sure that you first take a quick look at the
integrand and see if there are any continuity problems with the integrand and
if they occur in the interval of integration.
[Return to Problems]
(b)
Now, in this case the integral can be done because the two
points of discontinuity, ,
are both outside of the interval of integration. The substitution and converted limits in
this case are,
The integral is
then,
[Return to Problems]
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Let’s work another set of examples. These are a little tougher (at least in
appearance) than the previous sets.
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Example 4 Evaluate
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
(a)
The limits are a little unusual in this case, but that
will happen sometimes so don’t get too excited about it. Here is the substitution.
The integral is then,
[Return to Problems]
(b)
Here is the substitution and converted limits for this
problem.
The integral is,
[Return to Problems]
(c)
Here is the substitution and converted limits and don’t
get too excited about the substitution.
It’s a little messy in the case, but that can happen on occasion.
Here is the integral,
So, not only was the substitution messy, but we also a
messy answer, but again that’s life on occasion.
[Return to Problems]
(d)
This problem not as bad as it looks. Here is the substitution and converted
limits.
The cosine in the very front of the integrand will get
substituted away in the differential and so this integrand actually
simplifies down significantly. Here is
the integral.
Don’t get excited about these kinds of answers. On occasion we will end up with trig
function evaluations like this.
[Return to Problems]
(e)
This is also a tricky substitution (at least until you see
it). Here it is,
Here is the integral.
[Return to Problems]
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In this last set of examples we saw some tricky
substitutions and messy limits, but these are a fact of life with some
substitution problems and so we need to be prepared for dealing with them when
they happen.
Even and Odd
Functions
This is the last topic that we need to discuss in this
chapter. It is probably better suited in
the previous section, but that section has already gotten fairly large so I
decided to put it here.
First, recall that an even function is any function which
satisfies,
Typical examples of even functions are,
An odd function is any function which satisfies,
The typical examples of odd functions are,
There are a couple of nice facts about integrating even and
odd functions over the interval [-a,a]. If f(x)
is an even function then,
Likewise, if f(x)
is an odd function then,
Note that in order to use these facts the limit of
integration must be the same number, but opposite signs!
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Example 5 Integrate
each of the following.
(a) [Solution]
(b) [Solution]
Solution
Neither of these are terribly difficult integrals, but we
can use the facts on them anyway.
(a)
In this case the integrand is even and the interval is
correct so,
So, using the fact cut the evaluation in half (in essence
since one of the new limits was zero).
[Return to Problems]
(b)
The integrand in this case is odd and the interval is in
the correct form and so we don’t even need to integrate. Just use the fact.
[Return to Problems]
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Note that the limits of integration are important here. Take the last integral as an example. A small change to the limits will not give us
zero.
The moral here is to be careful and not misuse these facts.