Lesson 6 3 The Indefinite Integral 2 3

Lesson 6. 3 The Indefinite Integral

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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) 5

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 6

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

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

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

Substitution • Consider the problem of taking the integral of § We could expand the binomial (messy) • New strategy … substitute u = 4 x – 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 10

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

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

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

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 14

Assignment • Lesson 6. 3 • Page 234 • Exercises 1, 5, 9, … 69 (every other odd) 15