Worksheet for Section 6.2

Secton 6.2 is about antiderivatives, or what happens when you try to interpret the derivative rules from Chapter 2 backwards. For example, suppose you know that the function $F(x)$ has derivative $F'(x)=f(x)=3x^2$ -- then what is $F(x)$? In this case, $f(x)=3x^2$ is in a form that suggests the result of applying the Power Rule for derivatives, and in fact, with $F(x)=x^3$, $F'(x)=3x^2=f(x)$ as needed. However, note that also $F(x)=x^3+4$, $F(x)=x^3-12$, and any other function of the form $F(x)=x^3+C$ (where $C$ is a constant) will also satisfy $F'(x)=3x^2=f(x)$. In other words, the antiderivative $F(x)$ of the function $f(x)$ is not unique -- there are many possible antiderivatives, differing only by a constant term. In fact, if $F(x)$ is an antiderivative of $f(x)$, then another function $G(x)$ is also an antiderivative of $f(x)$ just in case $G(x)=F(x)+C$ for some constant $C$. The notation for this antiderivative situation is written as follows:

$$\int f(x)\, dx=F(x)+C$$

The constant of integration $C$ is an important part of this expression, making sure that you get a general solution for the antiderivative. All of the basic derivative rules from Chapter 2 have antiderivative forms -- these are listed on pp. 289-90 in the book, and also inside the back cover. Here are some examples of antiderivatives. In each case, give your solution in general form, with the constant of integration. In some cases, you may need to rewrite the expression a bit before applying the antiderivative rule.

$$\int 5\, dx\qquad\qquad \int 2x^2\, dx\qquad\qquad \int \frac{3}{x^2}\, dx$$

$$\int 3\cos x\, dx\qquad\qquad \int \frac{x-2}{\sqrt{x}}\, dx\qquad\qquad \int \frac{\sin x}{\cos^2 x}\, dx$$

The same sum-and-difference and constant multple properties that you saw in Section 5.4 for definite integrals apply also for antiderivatives here. This allows you to take antiderivatives of any polynomial using the Power Rule, for example. Also, you can now compute definite integrals using antiderivatives with the Fundamental Theorem. I will show you examples of these in class.