So far, we've focused on differentiation, the process of finding the rate of change of a function. For example, if we have the function $f(x) = x^3$, its derivative is $f'(x) = 3x^2$. But what if we go in the opposite direction? What if we start with the derivative, say $3x^2$, and want to find the original function?

This reverse process is called antidifferentiation. The function we find is called an antiderivative.

Let's stick with our example. We know that an antiderivative of $f(x) = 3x^2$ is $F(x) = x^3$ because the derivative of $x^3$ is $3x^2$. But is that the only one?

Think about the function $G(x) = x^3 + 5$. What's its derivative? It's also $3x^2$, since the derivative of a constant is zero. The same goes for $H(x) = x^3 - 100$. Its derivative is also $3x^2$.

It seems there are infinitely many antiderivatives for any given function, all differing by a constant. We account for this by adding a "constant of integration," which we represent with the letter $C$.

This family of functions includes all possible antiderivatives. The value of $C$ simply shifts the function's graph up or down the y-axis, without changing its shape or the slope of its tangent line at any point.

Finding an antiderivative is also called integration. The notation for finding the antiderivative of a function $f(x)$ is called an indefinite integral:

The symbol $\int$ is the integral sign. The function $f(x)$ is the integrand, and $dx$ indicates that we are integrating with respect to the variable $x$. Don't forget the $+ C$!

Just as we have rules for differentiation, we have rules for integration. Many are just the reverse of the differentiation rules we already know. The most fundamental one is the Power Rule for Integration.

Let's try it out. To find the integral of $x^4$, we add 1 to the exponent (making it 5) and divide by 5.

We can check this by taking the derivative. The derivative of $\frac{x^5}{5} + C$ is $\frac{1}{5} \cdot 5x^4 + 0$, which simplifies back to $x^4$. It works!

Here are a few more essential rules. They follow directly from their derivative counterparts.

Let's combine these rules to solve a more complex integral.

First, we can write the integral expression:

Using the Sum/Difference Rule, we can integrate each term separately:

Now, we apply the Constant Multiple Rule and the Power Rule to each term:

Finally, we simplify the expression. Notice we only need one $+ C$ at the end to represent the constant for the entire antiderivative.

Antidifferentiation is a fundamental skill in calculus. By reversing the process of differentiation, it opens the door to solving a whole new class of problems, like calculating the area under a curve or the distance traveled by an object.

Ready to check your understanding?

Great work. Mastering these basic rules is the first step toward understanding the full power of integration.