Substitution Rule for Indefinite Integrals
After the last section we now know how to do the following
integrals.
However, we can�t do the following integrals.
All of these look considerably more difficult than the
first set. However, they aren�t too bad once you see how to do them. Let�s
start with the first one.
In this case let�s notice that if we let
and we compute the
differential (you remember how to compute these right?) for this we get,
Now, let�s go back to our integral and notice that we can
eliminate every x that exists in the integral and write the integral
completely in terms of u using both the definition of u and its
differential.
In the process of doing this we�ve taken an integral that
looked very difficult and with a quick substitution we were able to rewrite the
integral into a very simple integral that we can do.
Evaluating the integral gives,
As always we can check our answer with a quick derivative
if we�d like to and don�t forget to �back substitute� and get the integral back
into terms of the original variable.
What we�ve done in the work above is called the
Substitution Rule. Here is the substitution rule in general.
Substitution Rule
A natural question at this stage is how to identify the
correct substitution. Unfortunately, the answer is it depends on the integral.
However, there is a general rule of thumb that will work for many of the
integrals that we�re going to be running across.
When faced with an integral we�ll ask ourselves what we
know how to integrate. With the integral above we can quickly recognize that we
know how to integrate
However, we didn�t have just the root we also had stuff in
front of the root and (more importantly in this case) stuff under the root.
Since we can only integrate roots if there is just an x under the root a
good first guess for the substitution is then to make u be the stuff
under the root.
Another way to think of this is to ask yourself what
portion of the integrand has an inside function and can you do the integral with
that inside function present. If you can�t then there is a pretty good chance
that the inside function will be the substitution.
We will have to be careful however. There are times when
using this general rule can get us in trouble or overly complicate the problem.
We�ll eventually see problems where there are more than one �inside function�
and/or integrals that will look very similar and yet use completely different
substitutions. The reality is that the only way to really learn how to do
substitutions is to just work lots of problems and eventually you�ll start to
get a feel for how these work and you�ll find it easier and easier to identify
the proper substitutions.
Now, with that out of the way we should ask the following
question. How, do we know if we got the correct substitution? Well, upon
computing the differential and actually performing the substitution every x
in the integral (including the x in the dx) must disappear in the
substitution process and the only letters left should be u�s (including a
du). If there are x�s left over then there is a pretty good
chance that we chose the wrong substitution. Unfortunately, however there is at
least one case (we�ll be seeing an example of this in the next section) where
the correct substitution will actually leave some x�s and we�ll need to
do a little more work to get it to work.
Again, it cannot be stressed enough at this point that the
only way to really learn how to do substitutions is to just work lots of
problems. There are lots of different kinds of problems and after working many
problems you�ll start to get a real feel for these problems and after you work
enough of them you�ll reach the point where you�ll be able to do simple
substitutions in your head without having to actually write anything down.
As a final note we should point out that often (in fact in
almost every case) the differential will not appear exactly in the integrand as
it did in the example above and sometimes we�ll need to do some manipulation of
the integrand and/or the differential to get all the x�s to disappear in
the substitution.
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