We’ve now seen a fair number of different integration
techniques and so we should probably pause at this point and talk a little bit
about a strategy to use for determining the correct technique to use when faced
with an integral.
There are a couple of points that need to be made about this
strategy. First, it isn’t a hard and
fast set of rules for determining the method that should be used. It is really nothing more than a general set
of guidelines that will help us to identify techniques that may work. Some integrals can be done in more than one
way and so depending on the path you take through the strategy you may end up
with a different technique than somebody else who also went through this
strategy.
Second, while the strategy is presented as a way to identify
the technique that could be used on an integral also keep in mind that, for
many integrals, it can also automatically exclude certain techniques as
well. When going through the strategy
keep two lists in mind. The first list
is integration techniques that simply won’t work and the second list is
techniques that look like they might work.
After going through the strategy and the second list has only one entry
then that is the technique to use. If,
on the other hand, there is more than one possible technique to use we will
then have to decide on which is liable to be the best for us to use. Unfortunately there is no way to teach which
technique is the best as that usually depends upon the person and which
technique they find to be the easiest.
Third, don’t forget that many integrals can be evaluated in
multiple ways and so more than one technique may be used on it. This has already been mentioned in each of
the previous points, but is important enough to warrant a separate
mention. Sometimes one technique will be
significantly easier than the others and so don’t just stop at the first
technique that appears to work. Always
identify all possible techniques and then go back and determine which you feel
will be the easiest for you to use.
Next, it’s entirely possible that you will need to use more
than one method to completely do an integral.
For instance a substitution may lead to using integration by parts or
partial fractions integral.
Finally, in my class I will accept any valid integration
technique as a solution. As already
noted there is often more than one way to do an integral and just because I
find one technique to be the easiest doesn’t mean that you will as well. So, in my class, there is no one right way of
doing an integral. You may use any
integration technique that I’ve taught you in this class or you learned in
Calculus I to evaluate integrals in this class.
In other words, always take the approach that you find to be the
easiest.
Note that this final point is more geared towards my class
and it’s completely possible that your instructor may not agree with this and
so be careful in applying this point if you aren’t in my class.
Okay, let’s get on with the strategy.
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- Simplify the integrand, if possible. This step is very important in the
integration process. Many
integrals can be taken from impossible or very difficult to very easy
with a little simplification or manipulation. Don’t forget basic trig and algebraic
identities as these can often be used to simplify the integral.
We used this idea when we were looking at integrals involving trig
functions. For example consider
the following integral.
This integral can’t be done as is, however simply by recalling the identity,
the integral becomes very easy to
do.
Note that this example also shows
that simplification does not necessarily mean that we’ll write the integrand
in a “simpler” form. It only means
that we’ll write the integrand into a form that we can deal with and this is
often longer and/or “messier” than the original integral.
- See if a “simple” substitution will
work. Look to see if a simple
substitution can be used instead of the often more complicated methods
from Calculus II. For example
consider both of the following integrals.
The first integral can be done
with partial fractions and the second could be done with a trig
substitution.
However, both could also be evaluated using the substitution and the work involved in the substitution
would be significantly less than the work involved in either partial
fractions or trig substitution.
So, always look for quick, simple substitutions before moving on to the more
complicated Calculus II techniques.
- Identify the type of integral. Note that any integral may fall into
more than one of these types.
Because of this fact it’s usually best to go all the way through
the list and identify all possible types since one may be easier than
the other and it’s entirely possible that the easier type is listed
lower in the list.
- Is
the integrand a rational expression (i.e is the integrand a polynomial divided by a
polynomial)? If so, then partial
fractions may work on the integral.
- Is
the integrand a polynomial times a trig function, exponential, or
logarithm? If so, then
integration by parts may work.
- Is
the integrand a product of sines and cosines, secant and tangents, or
cosecants and cotangents? If so,
then the topics from the second section may work.
Likewise, don’t forget that some quotients involving these functions
can also be done using these techniques.
- Does
the integrand involve ,
,
or ? If so, then a trig substitution might
work nicely.
- Does
the integrand have roots other than those listed above in it? If so, then the substitution might work.
- Does
the integrand have a quadratic in it?
If so, then completing the square on the quadratic might put it
into a form that we can deal with.
- Can we relate the integral to an
integral we already know how to do?
In other words, can we use a substitution or manipulation to
write the integrand into a form that does fit into the forms we’ve
looked at previously in this chapter.
A typical example here is the following integral.
This integral doesn’t obviously
fit into any of the forms we looked at in this chapter. However, with the substitution we can reduce the integral to the form,
which is a trig substitution
problem.
- Do we need to use multiple
techniques? In this step we
need to ask ourselves if it is possible that we’ll need to use multiple
techniques. The example in the
previous part is a good example.
Using a substitution didn’t allow us to actually do the
integral. All it did was put the
integral and put it into a form that we could use a different technique
on.
Don’t ever get locked into the idea that an integral will only require
one step to completely evaluate it.
Many will require more than one step.
- Try again. If everything that you’ve tried to
this point doesn’t work then go back through the process and try
again. This time try a technique
that that you didn’t use the first time around.
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As noted above this strategy is not a hard and fast set of
rules. It is only intended to guide you
through the process of best determining how to do any given integral. Note as well that the only place Calculus II
actually arises is in the third step.
Steps 1, 2 and 4 involve nothing more than manipulation of the integrand
either through direct manipulation of the integrand or by using a
substitution. The last two steps are
simply ideas to think about in going through this strategy.
Many students go through this process and concentrate almost
exclusively on Step 3 (after all this is Calculus II, so it’s easy to see why
they might do that….) to the exclusion of the other steps. One very large consequence of that exclusion
is that often a simple manipulation or substitution is overlooked that could
make the integral very easy to do.
Before moving on to the next section we should work a couple of quick problems
illustrating a couple of not so obvious simplifications/manipulations and a not
so obvious substitution.
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Example 1 Evaluate
the following integral.
Solution
This integral almost falls into the form given in 3c.
It is a quotient of tangent and secant and we know that sometimes we
can use the same methods for products of tangents and secants on
quotients.
The process from that section
tells us that if we have even powers of secant to strip two of them off and
convert the rest to tangents. That
won’t work here. We can split two
secants off, but they would be in the denominator and they won’t do us any
good there. Remember that the point of
splitting them off is so they would be there for the substitution .
That requires them to be in the numerator. So, that won’t work and so we’ll have to
find another solution method.
There are in fact two solution methods to this integral
depending on how you want to go about it.
We’ll take a look at both.
Solution 1
In this solution method we could just convert everything
to sines and cosines and see if that gives us an integral we can deal with.
Note that just converting to sines and cosines won’t
always work and if it does it won’t always work this nicely. Often there will be a lot more work that
would need to be done to complete the integral.
Solution 2
This solution method goes back to dealing with secants and
tangents. Let’s notice that if we had
a secant in the numerator we could just use as a substitution and it would be a fairly
quick and simple substitution to use.
We don’t have a secant in the numerator. However we could very easily get a secant
in the numerator simply by multiplying the numerator and denominator by
secant.
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In the previous example we saw two “simplifications” that
allowed us to do the integral. The first
was using identities to rewrite the integral into terms we could deal with and
the second involved multiplying the numerator and the denominator by something
to again put the integral into terms we could deal with.
Using identities to rewrite an integral is an important
“simplification” and we should not forget about it. Integrals can often be greatly simplified or
at least put into a form that can be dealt with by using an identity.
The second “simplification” is not used as often, but does
show up on occasion so again, it’s best to not forget about it. In fact, let’s take another look at an
example in which multiplying the numerator and denominator by something will
allow us to do an integral.
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Example 2 Evaluate
the following integral.
Solution
This is an integral in which if we just concentrate on the
third step we won’t get anywhere. This
integral doesn’t appear to be any of the kinds of integrals that we worked in
this chapter.
We can do the integral however, if we do the following,
This does not appear to have done anything for us. However, if we now remember the first
“simplification” we looked at above we will notice that we can use an
identity to rewrite the denominator.
Once we do that we can further reduce the integral into something we
can deal with.
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So, we’ve seen once again that multiplying the numerator and
denominator by something can put the integral into a form that we can
integrate. Notice as well that this
example also showed that “simplifications” do not necessarily put an integral
into a simpler form. They only put the
integral into a form that is easier to integrate.
Let’s now take a quick look at an example of a substitution
that is not so obvious.
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Example 3 Evaluate
the following integral.
Solution
We introduced this example saying that the substitution
was not so obvious. However, this is
really an integral that falls into the form given by 3e in our strategy
above. However, many people miss that
form and so don’t think about it. So,
let’s try the following substitution.
With this substitution the integral becomes,
This is now an integration by parts integral. Remember that often we will need to use
more than one technique to completely do the integral. This is a fairly simple integration by
parts problem so I’ll leave the remainder of the details to you to check.
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Before leaving this section we should also point out that
there are integrals out there in the world that just can’t be done in terms of
functions that we know. Some examples of
these are.
That doesn’t mean that these integrals can’t be done at some
level. If you go to a computer algebra
system such as Maple or Mathematica and have it do these integrals it will return the following.
So it appears that these integrals can in fact be done. However this is a little misleading. Here are the definitions of each of the
functions given above.
Error Function
The Sine Integral
The Fresnel Cosine
Integral
The Cosine Integral
Where is the Euler-Mascheroni constant.
Note that the first three are simply defined in terms of
themselves and so when we say we can integrate them all we are really doing is
renaming the integral. The fourth one is
a little different and yet it is still defined in terms of an integral that
can’t be done in practice.
It will be possible to integrate every integral given in
this class, but it is important to note that there are integrals that just
can’t be done. We should also note that
after we look at Series we will be able to write
down series representations of each of the integrals above.