Substitution is the most powerful and at the same time perhaps the easiest method of evaluating integrals, which is why it is our method of choice. It has one disadvantage: sometimes it fails. Therefore we not only need to learn the mechanics of substitution itself, but also acquire some feeling for when the substitution works and when it fails. This comes with experience. The general idea of sustitution (not just in integral) is described in this note.
is based on this mathematical theorem. In the ideal case it can be expressed by this procedure:
That is, we choose a transformation
If this substitution succeeds, we obtain a new integral and with a bit of luck we can solve it. Then one has to also do a so-called back substitution, that is, return back to the original variable, but that is simple, we have a formula for it.
How does this tie in with the formal theorem on substitution? We have a
The beauty of the procedure we use above above is that it allows us to
dispose of all this theoretical reasoning, we do not worry about J
and I, we simply want to exchange one expression for another.
We also need not remember that according to that theorem we also need to
have the derivative of g in that complicated integral with composed
f. The substituting procedure forces us to replace not dx but
The main weakness is that we need to have that
The basic requirement is that after the substitution, all
x must disappear from the integral. The only tool available is the
By this we mean any substitution of the form
The integral is of course only valid for
is based on the following mathematical theorem. In the ideal case it can be expressed by this procedure:
It looks as if we followed the above direct substitution in the opposite direction and there is something to it. There we tried to simplify the integral by replacing a (complicated) expression with a single letter, here we make the integral more complicated, but the basic mechanism stays the same: We choose a transformation and then use it to change the integral, we also get an equality for changing the defferential int he same way, by taking derivative of g and using it as a coefficient. However, deep inside it is a different procedure, which can be seen for instance from the fact that here requirements on g are more strict (in particular, it must be invertible, unlike the situation in direct substitution).
There is also a difference in difficulty of various stages. The direct
substitution tight at the start, especially with replacing the dx
which often makes the chosen substitution impossible, but the back
substitution is clear. On the other hand, the indirect substitution has no
trouble with dx, from
Now why would anyone want to make an integral more complicated then it already is? Not surprisingly, the indirect substitution is used quite rarely, but there are specific types of integrals where it does help and experienced integrators know beforehand that certain indirect substitutions will in the end magically simplify (see integrals with roots in Methods Survey - Integration). We will show a very simple example of this kind.
Actually, this is an elementary integral that one should remember.
Note that even this simple example is already good enough to
illustrate that the indirect substitution is not as simple as it looks. A
careful reader should get mightily suspicious at the first integral in the
second line: Shouldn't there be an absolute value in the denominator? That
is a very good question and the answer depends heavily on how we actually do
this indirect substitution. According to the theorem we are supposed to take
is a combination of direct and indirect substitution. We use it to shorten
calculations when we have to use several substitutions in a row. Experienced
integrators can save quite a lot of time this way. It starts with a
Also here one has to be careful on which interval the new variable lives, in
this particular example one has to decide between intervals
This is basically the same as using substitution in an indefinite integral.
We state that the basic rule in substitution is to change everything from
the language of x to the language of y, and that include also
the limits. How do we transform them? As usual, using our basic
Similarly we change limits in the indirect and mixed substitution, but there
it is somewhat more difficult. For instance, we had a substitution
For practical hints and examples see Substitution in Methods Survey - Methods of integration.
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