In this section, we introduce the idea of the indefinite integral. We also look at the process of variable substitution to find antiderivatives of more complex functions.

For any function f, is called the indefinite integral of f and represents the most general antiderivative of f.

For any function f, is called the indefinite integral of f and represents the most general antiderivative of f. For example:

Find each of the following: (a) (b) (c)

Find each of the following: (d) (e) (f)

What if the integrand is not something that we recognize as a “basic” antiderivative rule? For example, We would like to do a variable substitution so that the “new” integral is one that we recognize as a “basic” antiderivative rule.

Let f, u, and g be continuous functions such that: for all x. Then:

Substitute: Choose a function u = u(x) such that the substitution of u for x and du for dx changes into Antidifferentiate: Solve - that is, find G(u) such that Resubstitute: Substitute x back in for u to get the answer to have an antiderivative of the original function

Looking again at

Find each of the following: (a) (b)

Find each of the following: (a) (b)

Find each of the following: (a) (b)

Let f, u, and g be continuous functions such that: for all x. Then:

Find each of the following: (a) (b)