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
*y* = *g*(*x*)*g*(*x*)*y*. We also need to replace *dx*. For this we use the equality we
deduced in the second row, according to it we replace
*g*'(*x*)*dx**dy*.

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
function
*f*(*y*) = *e*^{y}*f* composed with
*g*(*x*) = sin(*x*).*J* of all real
numbers, since we always want the integral on the largest possible set and
there is no reason to restrict ourselves to something smaller. The function
*g* maps this interval onto the interval
*I* = [-1,1].*f* on the interval *I*, which is the middle part in the
calculation above. Into the formula that we obtained we substitute *g*
and get the answer to our original question, valid on the original
interval *I*.

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
*g*'(*x*)*dx*.

The main weakness is that we need to have that
*g*'(*x*)*dx* and that *x*
must appear exclusively in the form of *g*(*x*),*g* would not be possible.
However, this is rarely possible. Fortunately, one can help it, which makes
substitution one of the most powerful tools. How does it work?

The basic requirement is that after the substitution, all
*x* must disappear from the integral. The only tool available is the
chosen transformation *y* = *g*(*x*)*y* = *g*'(*x*)*dx*,*x* with
expressions featuring *y*. This gives us tremendous freedom, but one
should remember that in such a case we do not use the quoted theorem as
stated, so then we are on a shaky ground and the answer may come out wrong.
It is therefore strongly recommended to check that our result is correct (by
differentiating it) - actually, we should do this with every integral we
solve. Actually, there is rarely any trouble unless you get really wild with
those formulas. We will now show one example of this more general approach.

**Linear substitution.**

By this we mean any substitution of the form
*y* = *Ax* + *B*,*dy* = *A*⋅*dx*.*A*⋅*dx**dx* = (1/*A*)*dy**dx* in the integral. This means that we can always do any linear
substitution. We will show one fairly important example.

The integral is of course only valid for
*a* > 0,

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
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
*x* = *g*(*t*)*x* = *g*'(*t*)*dt*,

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
a function *g* = sin(*t*)*J* to our interval *J* must be chosen in such a way that *g* is 1-1 there and the
image of *J* under *g* must be the whole interval
*J*, then we might be forced to use the absolute value.
As we see, this is another difference, in direct substitution we need not
worry about intervals used in the procedure.

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
transformation
*h*(*y*) = *g*(*x*)*h*'(*y*)*dy* = *g*'(*x*)*dx*.

Also here one has to be careful on which interval the new variable lives, in
this particular example one has to decide between intervals
*y*^{2}

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
transformation
*y* = *g*(*x*).

Similarly we change limits in the indirect and mixed substitution, but there
it is somewhat more difficult. For instance, we had a substitution
*y*^{2} = *x*^{3} + 8*x* = 2,*y*. In this
case there are two possibilities, plus and minus 4, the right one is
determined by the choice of the interval *J* in the substitution.

For practical hints and examples see Substitution in Methods Survey - Methods of integration.

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