1. The Indefinite Integral
Lesson 6.3
The Indefinite Integral

5. Review Fundamental Theorem of Calculus said
We called this the definite integral
We used the antiderivative of f(x)
The general form is called the indefinite integral
Used to indicate all the antiderivatives of f(x)

6. Indefinite Integral The definite integral is a number
The indefinite integral is a family of functions
The +C represents an arbitrary constant
The constant of integration

7. Indefinite Integral
Consider the family of antiderivatives of the function y = sin x

8. Properties of Indefinite Integrals
The derivative of the indefinite integral is the original function
A constant can be factored out of the integral

9. Properties of Indefinite Integrals
The integral of a sum (difference) is the sum (difference) of the integrals
The antiderivative rule (power rule)

10. Substitution Consider the problem of taking the integral of
We could expand the binomial (messy)
New strategy … substitute u = 4x – 6
What is the derivative of u with respect to x?
Now we make the substitution
The ¼ adjusts for the 4 in the du

11. Substitution The resulting integral is much simpler
Now we reverse the substitution and simplify

12. Try Another What will we substitute … u = ? What is the du ?
Now rewrite the integral and proceed

13. How About Another? Consider u = ? du = ? u = x2 + 5 du = 2x dx
Problem …
2x is not a constant
Cannot adjust the integral with a non constant coefficient
Substitution will not work for this integral

14. Application
We are told that a certain bacteria population is increasing a rate of
What is the increase in the population during the first 8 hours

15. Assignment Lesson 6.3 Page 234
Exercises 1, 5, 9, … 69 (every other odd)