Substitution is a general method used to simplify complicated expressions by assigning a simple name to some part of that expression.
Example: Consider the function
If we decide to denote
As you can see, substitution is done by simple replacement. This is a general rule. There are two basic rules for using substitution.
1. When doing a substitution, the original variable must completely
disappear from the given expression.
2. When replacing appearances of the original variable with the new
one, we can only use the basic substitution equality and information derived
from it.
For instance, in the above example we might be tempted to use the
substitution
Thus we get
More complicated expressions are of course possible, but whatever formulas are used for replacement must be derived from the original substitution equality. We can use algebra, but we have to be careful. Consider the following two examples.
Examples:
The first substitution shows that we can also pass to inverse functions when working with the substitution equality. The second substitution shows that we have to be careful when we do it.
Generally, when we do a substitution, an important part is for what values of
variables it is done. In the example at the beginning we did not care very
much, but now we have to. In the logarithmic substitution we have to make
sure that the logarithm exists, which means that the substitution can be done
only for
However, this is not always possible, as we see in the case of the cosine
substitution. There we need to restrict x to some set on which the
cosine is 1-1. Typically we do substitution on an interval, so here we have a
natural candidate: x should be taken from
But even if we did not have to pass to the inverse, there is yet another
trouble in that substitution. The formula for sine with square root works
only under the assumption that the sine is not negative. If we indeed
restrict
x to
Although most substitutions do not cause such troubles, sometimes one has to be very careful about where calculations are made.
When we do substitution in a limit, we have to change all appearances of the original variable, which also includes the place under "lim". There we simply change the name of the variable, but the limit point itself must change according to the basic substitution equality.
Example:
When we want to differentiate a function
Then we have to change back into the original variable. In fact, this is just a restatement of the chain rule, you can find more information there.
Example: Consider the function
If we want to find its derivative, we may do it directly, but then we would have to use the chain rule at quite a few places. It might be easier to first simplify the expression using substitution:
Now we easily calculate
Finally we use the rule for derivative with substitution:
There are three places related to the setting where the original variable can
appear. It is always present in the
Concerning the differential, we have a simple rule:
The substitution
Example:
Since integration is rather specific, we refer to substitution in Integrals - Theory - Methods for more information.
When changing the indexing variable in a sum (finite or infinite), we also have to change its limits. It might also be necessary to switch the limits, since unlike integrals, the order of limits does not have any real meaning and we always put the smaller down.
Example: