Antiderivatives and Indefinite Integrals

1. Reversing Differentiation: The Antiderivative

Throughout the last two modules, we've focused on one question: given a function, what is its derivative (its rate of change)?

Now, let's flip that question around: Given a derivative, what was the original function?

For example, if we know that the derivative of some function $F(x)$ is $f(x)=2x$, can we figure out what $F(x)$ is?
You might correctly guess that $F(x)=x^2$, since $\frac{d}{dx}(x^2)=2x$.

But what about the function $F(x)=x^2+5$? Its derivative is also $\frac{d}{dx}(x^2+5)=2x+0=2x$.
And what about $F(x)=x^2-100$? Its derivative is also $2x$.

It seems there is a whole family of functions whose derivative is $2x$. They all have the form $x^2+C$, where $C$ is any constant number. This is because the derivative of any constant is zero.

This leads us to our first key definition:

A function $F(x)$ is called an antiderivative of a function $f(x)$ if $F^{\prime }(x)=f(x)$.

To see this concept explained visually, please watch the beginning of the following video.

This idea of a family of antiderivatives, all differing by a constant, is crucial. The constant is often called the constant of integration.

2. The Indefinite Integral

Mathematicians developed a specific notation to represent the operation of finding all antiderivatives of a function. This is called the indefinite integral.

The indefinite integral of a function $f(x)$ is written as:
$\int f(x)dx=F(x)+C$

Let's break down this notation.

• The $\int$ symbol is the integral sign. It's an elongated "S," which stands for "sum." The reason for this will become clear in a few lessons when we discuss definite integrals and area.
• The function $f(x)$ is called the integrand.
• The $dx$ is the differential of x. It specifies that we are integrating with respect to the variable $x$.
• The result, $F(x)+C$, is the general antiderivative of $f(x)$.

So, "finding the indefinite integral" is just a formal way of saying "finding the general antiderivative." You might also see an antiderivative referred to as a primitive function.

For a more formal definition and summary, please read the highlighted sections of the following guide.

3. Finding Basic Integrals: The Power Rule in Reverse

Now that we understand what an indefinite integral is, how do we find one? We can start by reversing the rules we learned for differentiation.

Let's recall the power rule for differentiation:
To differentiate $x^n$, you:

1. Multiply by the power $n$.
2. Subtract 1 from the power.

To find the integral $\int x^{n}dx$, we must perform the opposite operations in the reverse order:

1. Add 1 to the power (to get $n+1$).
2. Divide by the new power ($n+1$).

This gives us the power rule for integration:
$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C(\text{for }n\ne -1)$
The case where $n=-1$ (i.e., $\int \frac{1}{x}dx$) is special and we will cover it in a later module.

This video does an excellent job of demonstrating this reversal process.

Let's work through an example together. Find $\int (3x^2+8x-5)dx$.

Just like with differentiation, we can integrate term by term.

1. Integrate $3x^2$:

   • Add 1 to the power: $2+1=3$.
   • Divide by the new power: $\frac{3x^3}{3}=x^3$.

2. Integrate $8x$: (Remember $x$ is $x^1$)

   • Add 1 to the power: $1+1=2$.
   • Divide by the new power: $\frac{8x^2}{2}=4x^2$.

3. Integrate $-5$: (Think of $-5$ as $-5x^0$)

   • Add 1 to the power: $0+1=1$.
   • Divide by the new power: $\frac{-5x^1}{1}=-5x$.

4. Combine and add the constant:
   We only need to add one constant of integration at the very end to represent the whole family of antiderivatives.
   $\int (3x^2+8x-5)dx=x^3+4x^2-5x+C$
   You can always check your answer by differentiating it: $\frac{d}{dx}(x^3+4x^2-5x+C)=3x^2+8x-5$. It matches our original integrand, so we know our answer is correct.

Conclusion

Congratulations on taking your first step into integral calculus! This lesson introduced the conceptual foundation for the rest of this module.

Key Takeaways:

• Antidifferentiation is the process of reversing differentiation: finding a function from its derivative.
• An antiderivative of $f(x)$ is a function $F(x)$ such that $F^{\prime }(x)=f(x)$.
• The indefinite integral, $\int f(x)dx$, represents the entire family of antiderivatives, which we write as $F(x)+C$.
• The + C is the constant of integration, and it's essential because the derivative of any constant is zero.
• We can begin finding integrals by reversing the differentiation rules we already know, such as the power rule.

In our next lesson, we will build on this foundation. We'll establish a table of basic integrals (or antiderivative formulae/formulas) for polynomials, and some trigonometric, exponential, and logarithmic functions, giving us the tools to integrate a wider variety of expressions.