1. The Indefinite Integral
Objective: Develop some fundamental results about antidifferentiation

2. Definition Definition 5.2.1
A function F is called an antiderivative of a function f on a given interval I if F/(x) = f(x) for all x in the interval.

3. Definition Definition 5.2.1
A function F is called an antiderivative of a function f on a given interval I if F/(x) = f(x) for all x in the interval.
For example, the function is an antiderivative of on the interval
because for each x in this interval

4. Definition
However, is not the only antiderivative of f on this interval. If we add any constant C to
this would also be an antiderivative. We will express this as

5. Theorem 5.2.2
If F(x) is any antiderivative of f(x) on an interval I, then for any constant C the function F(x) + C is also an antiderivative on that interval. Moreover, each antiderivative of f(x) on the interval I can be expressed in the form F(x) + C by choosing the constant C appropriately.

6. The Indefinite Integral
The process of finding antiderivatives is called antidifferentiation or integration. Thus, if
then integrating the function f(x) produces an antiderivative of the form F(x) + C. This is written

7. The Indefinite Integral
The process of finding antiderivatives is called antidifferentiation or integration. Thus, if
then integrating the function f(x) produces an antiderivative of the form F(x) + C. This is written
Note that if we differentiate an antiderivative of f(x), we obtain f(x) again. Thus

8. The Indefinite Integral
The differential symbol dx, in the differentiation and antidifferentiation operations
serves to identify the independent variable. If an independent variable other than x is used, then we would change the notation appropriately.

9. The Power Rule in Reverse
Lets look at the Power Rule again. We are going to do everything in reverse.
Differentiation Integration
Mult. by the exponent Add 1 to the exponent
Subt. 1 from exponent Div by the new exponent

10. The Power Rule in Reverse
Lets look at the Power Rule again. We are going to do everything in reverse.
Differentiation Integration
Mult. by the exponent Add 1 to the exponent
Subt. 1 from exponent Div by the new exponent

11. The Power Rule in Reverse
Here are some more examples. This process needs to be memorized.

12. Integration formulas
Here are some examples of derivative formulas and their equivalent integration formulas. These need to be memorized. You can find them on page 357.

13. Integration formulas
Here are some examples of derivative formulas and their equivalent integration formulas. These need to be memorized. You can find them on page 357.

14. Properties of the Indefinite Integral
Theorem 5.2.3
Suppose that F(x) and G(x) are antiderivatives of f(x) and g(x) respectively, and that c is a constant. Then:
A constant factor can be moved through an integral sign.

15. Properties of the Indefinite Integral
Theorem 5.2.3
Suppose that F(x) and G(x) are antiderivatives of f(x) and g(x) respectively, and that c is a constant. Then:
b) An antiderivative of a sum is the sum of the antiderivative.

16. Properties of the Indefinite Integral
Theorem 5.2.3
Suppose that F(x) and G(x) are antiderivatives of f(x) and g(x) respectively, and that c is a constant. Then:
b) An antiderivative of a diff. is the diff of the antiderivative.

17. Properties of the Indefinite Integral
Theorem can be summarized by the following formulas.

18. Example 2
Evaluate

19. Example 2
Evaluate

20. Example 2
Evaluate

21. Example 2
Evaluate

22. Example 3
Evaluate

23. Example 3
Evaluate

24. Example 4
Evaluate

25. Example 4
Evaluate

26. Example 4
Evaluate

27. Example 4
Evaluate

28. Example 4
Evaluate

29. Example 4
Evaluate

30. Integral Curves
Graphs of antiderivatives of a function are called integral curves of f. For example, is one integral curve for All other integral curves have equations of the form

31. Example 5
Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).

32. Example 5
Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).
“The tangent line has slope x2” means that is the value of the derivative of the function. To find the function, we integrate.

33. Example 5
Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).
“The tangent line has slope x2” means that is the value of the derivative of the function. To find the function, we integrate.

34. Example 5
Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).
Now we use the point (2, 1) to solve for C.

35. Example 5
Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).
Now we use the point (2, 1) to solve for C.

36. Homework
Section 5.2
1-35 odd