L&T, 19.3
This is the integral equivalent of the chain rule. If and then the chain rule says, . We can rearrange this “by multiplying by” to get,
. (This can be proven from the rule for finite steps,
which can be rearranged as
In the limit that goes to zero, as it must in the integral, we find the required result). This is the basic formula we need to convert an integral with respect to a new variable . It is true as a substitution rule inside the integral, not as a general equality.
Replace some function of by .
Example 5.14:
Evaluate .
Solution:
Substitute (try this), then . We can only use this substitution if we can identify as part of . To that end write . We can now substitute for and for , and thus , where the limits still need to be filled in. Since is now an integral w.r.t. , the limits must be starting and finishing values of . At the start, where , . At the finish , , so
Note: The integrand, (i.e., the object being integrated) changes from to . Part of this change is due to the change from to .
Note: The integration limits change (for definite integrals only).
Example 5.15:
Calculate the indefinite integral .
Solution:
Use substitution, and take , , .
Finally we must substitute back using ,
Several standard integrals can be generalised using this substitution (left as exercise).
Example 5.16:
Evaluate
Solution:
Using the substitution we find
Thus
Replace by a function of . Sometimes, instead of putting
(5.1) |
e.g., , we replace directly by putting
(5.2) |
This is really same as using ( 5.1 ) since we can rearrange this equation, (i.e., solve for ) to get ( 5.2 ). However, we can work directly from ( 5.2 ) by calculating . We then use the formula
(Remember that we also must change limits on a definite integral!)
Example 5.17:
Evaluate .
Solution:
Put , , . The limits change, , . We obtain