Section 5.2

 means find all the functions that when differentiated yield an answer of  3x².

In other words,

There are many possible answers:
x³ + 7
x³ + 1.5
x³ – 13
etc.

Each of these functions is called an antiderivative of  3x².   They can all be written in the form x³ + C, where C is some constant.  So we write

A synonym for antiderivative is indefinite integral and the process of finding an antiderivative is called integration.  Every derivative formula we have learned can be turned around to obtain a corresponding antiderivative formula.  For example, in section 3.3 we learned the following derivative rules:

In our new notation, they can be rewritten as antiderivative rules:

Make sure you know these three plus the rest of the antiderivative formulas in Table 5.2.1 on the bottom of p.324.

When you are given an antiderivative in the homework that cannot be done using one of the formulas in the table, do some algebra or trig to it to make it look like one of the formulas in the table.

Example:

Solution:  First divide the numerator by x to get:

Theorem 5.2.3b allows us to break this up into three integrals:

Using line 2 of Table 5.2.1 on the first integral, line 1 and theorem 5.2.3a on the second integral, and line 11 on the last integral gives us the answer of