.Example 19 __
Find .
SOLUTION:.
Here we have let and noted that
where
.The limit on the right in the starred equation is, by definition, the derivative of
F(x), that is, f(x).
Example 20 __
Reexpress , in terms of u if .
SOLUTION: When , and 2u du = dx. The limits of the given
integral are values of x. When we write the new integral in terms of the variable
u, then the limits, if written, must be the values of u that correspond to the given
limits. Thus, when x = 3, u = 1, and when x = 6, u = 2. Then
.Example 21 __
If gā² is continuous, find .
SOLUTION:
.Note that the expanded limit is, by definition, the derivative of g(x) at c.