Learning Outcomes

Upon completing this lesson, you will be able to:

• Apply the basic rules for finding antiderivatives, including the constant, constant multiple, and power rules.
• Use the Sum and Difference Rules to find the indefinite integral of polynomial functions.
• Recognize the special integration case for the function $f(x) = \frac{1}{x}$.
• Combine the basic rules to solve a variety of indefinite integral problems.

Topics Covered

• The Constant Rule
• The Power Rule
• Constant Multiple Rule
• Sum and Difference Rule

I. The Constant and Constant Multiple Rules

Just as with differentiation, we have a set of rules to make finding indefinite integrals (antiderivatives) a more efficient process than simply guessing and checking. The simplest rules deal with constants.

A. The Constant Rule

The integral of a constant $k$ is the linear function $kx$, plus the constant of integration $C$.

$$ \int k \cdot dx = kx + C $$

This is because the derivative of $kx + C$ is $\frac{d}{dx}(kx + C) = k + 0 = k$.

B. The Constant Multiple Rule

The integral of a constant multiplied by a function is the constant multiplied by the integral of the function. This rule allows us to "pull out" constant factors from the integral.

$$ \int k \cdot f(x) \cdot dx = k \int f(x) \cdot dx $$

Example 1

Evaluate

$$ \int 7 \cdot dx $$

Using the Constant Rule, where $k=7$, the integral is

$$ 7x + C $$

Example 2

Evaluate

$$ \int 5x^2 \cdot dx $$

Using the Constant Multiple Rule, we can pull the $5$ out:

$$ 5 \int x^2 \cdot dx $$

II. The Power Rule for Integration

The Power Rule is the most fundamental rule for integrating functions of the form $x^n$. It is the direct reverse of the Power Rule for differentiation.

To integrate $x^n$, where $n$ is any real number except $-1$, we add one to the exponent and then divide by the new exponent.

$$ \int x^n \cdot dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1 $$

The Special Case: $n = -1$ The Power Rule fails when $n=-1$ because it would result in division by zero.

The integral of $x^{-1}$ or $\frac{1}{x}$ is a special case that we learned in the previous chapter:

$$ \int x^{-1} \cdot dx = \int \frac{1}{x} \cdot dx = \ln|x| + C $$

Example 3

Evaluate

$$ \int x^4 \cdot dx $$

Here, $n=4$.

We add $1$ to the exponent ($4+1=5$) and divide by $5$

$$ \frac{x^5}{5} + C $$

Example 4

Evaluate

$$ \int \sqrt{x} \cdot dx $$

Rewrite with a rational exponent: $\int x^{1/2} \cdot dx$

Here, $n = 1/2$. The new exponent is $n+1 = \frac{1}{2} + 1 = \frac{3}{2}$

$$ \frac{x^{3/2}}{3/2} + C \text{ or } \frac{2}{3}x^{3/2} + C $$