Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

#6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Just like how there are basic rules for calculating derivatives, there are rules for calculating antiderivatives. Since antiderivatives are the inverses of derivatives, these rules are mostly the reverse of the basic derivative rules. 💡

#📈 Indefinite Integrals: Notation

Let’s first talk about a family of functions before we dive into reversing the derivative process.

Imagine we have two different antiderivatives, $F(x) = x^2+3$ and $G(x) = x^2-2$.

If we were to take the derivative of both of these functions, we would find that they both have the same derivative, 2x. If we reverse the derivative process through integration, how do we account for arriving at these two different antiderivatives? Introducing the magical constant $C$! 🪄

When we integrate 2x, the antiderivative is 2x+C where $C$ is any constant. This result is often referred to as a family of functions because they vary only in the value of their constants and all share the same derivative.

This type of integral is referred to as an indefinite integral because we can’t be sure which member of the family of antiderivatives is at play. If the bounds of the integral are not specified as they are in a definite integral, always add ‘+C’ to the end of your antiderivative!

Here’s a general look at the notation!

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

Where $F'(x)=f(x)$ and $C$ represents the integration constant.

#📏 Indefinite Integrals: Basic Rules

Now let’s look at how to reverse the process of some of the derivatives we learned early in our study of calculus.

#Reverse Power Rule

First up, we have the reverse power rule. This essentially refers to how to take the indefinite integral of a function, which is the reverse of the power rule used for differentiation. Suppose we have the following function:

$$f(x) = \frac{x^{n+1}}{n+1}+C$$

Where $n \neq -1$ since $n=-1$ causes $f(x)$ to be undefined.

What is its derivative?

If we recall the power rule for derivatives, we see that the derivative of $f(x)$ is ...