1. u-du: Integrating Composite Functions
AP Calculus

2. Integrating Composite Functions
(Chain Rule)
Remember: Derivatives Rules
Remember: Layman’s Description of Antiderivatives
*2nd meaning of “du” du is the derivative of an implicit “u”

3. Integrating Composite Functions
u-du Substitution
Integrating Composite Functions
(Chain Rule)
Revisit the Chain Rule
If let u = inside function
du = derivative of the inside
becomes

4. Development
from the layman’s idea of antiderivative
“The Family of functions that has the given derivative”
must have the derivative of
the inside in order to find
the antiderivative of the outside

5. A Visual Aid
USING u-du Substitution  a Visual Aid
REM: u = inside function
du = derivative of the inside
let u =
becomes now only working with f , the outside function

6. Working With Constants: Constant Property of Integration
With u-du Substitution
REM: u = inside function
du = derivative of the inside
Missing Constant?
u =
du =
Worksheet - Part 1

7. Example 1 : du given
Ex 1:

8. Example 2: du given
Ex 2:

9. Example 3: du given
Ex 3:

10. Example 4: du given
Ex 4:
TWO WAYS! Differ by a constant

11. Example 5: Regular Method

12. Working with Constants < multiplying by one>
Constant Property of Integration
ILL let u =
du = and
becomes =
Or alternately = =

13. Example 6 : Introduce a Constant - my method

14. Example 7 : Introduce a Constant

15. Example 8 : Introduce a Constant << triple chain>>

16. Example 9 : Introduce a Constant - extra constant

17. Example 10 : Polynomial

18. Example 11: Separate the numerator

19. Formal Change of Variables << the Extra “x”>>
Solve for x in terms of u
ILL: Let
Solve for x in terms of u then
and
becomes

20. Formal Change of Variables << the Extra “x”>>
Rewrite in terms of u - du

21. Formal Change of Variables << the Extra “x”>>
Solve for x in terms of u - du
<<alt. Method>>
- could divide or multiply by

22. Complete Change of Variables << Changing du >>
At times it is required to even change the du as the u is changed above.
We will solve this later in the course.

23. Development
must have the derivative of
the inside in order to find
the antiderivative of the outside
*2nd meaning of “dx”
dx is the derivative of an implicit “x” more later
if x = f then dx = f /