So far, we've focused on differentiation, the process of finding a function's rate of change. If you have a function describing the distance a car has traveled, its derivative tells you the car's speed at any given moment.

Now, let's flip that idea around. What if you know the car's speed at every moment and want to find the total distance traveled? You're not looking for a rate of change; you're looking for an accumulation of change. This reverse process is called integration.

The process of finding a function whose derivative is the function we started with is called finding the antiderivative. This is the core of integration.

An indefinite integral is the general form of the antiderivative. The notation looks like this:

Let's break that down:

• The $\int$ symbol is the integral sign. It looks like a stretched-out 'S' for "sum."
• $f(x)$ is the function you're integrating, called the integrand.
• $dx$ tells us that we are integrating with respect to the variable $x$.
• $F(x)$ is the antiderivative, meaning that $F'(x) = f(x)$.

But what about the $+ C$? This is the constant of integration. Think about it: the derivative of $x^2$ is $2x$. The derivative of $x^2 + 5$ is also $2x$. The derivative of $x^2 - 100$ is still $2x$. Because the derivative of any constant is zero, when we go in reverse, we lose that information. The $+ C$ acknowledges this unknown constant.

Just as we have rules for differentiation, we have rules for integration. The simplest one is the power rule, in reverse. To integrate $x^n$, you add one to the exponent and then divide by the new exponent.

While indefinite integrals give us a general function, definite integrals give us a specific number. This number often represents a total accumulation, like the total distance traveled or, most famously, the area under a curve between two points.

The notation adds boundaries, called limits of integration, to the integral sign:

This expression means "the integral of $f(x)$ from $a$ to $b$." It calculates the net signed area between the function's graph and the x-axis, from the vertical line $x=a$ to the vertical line $x=b$. Area above the x-axis is positive, and area below it is negative.

So how do we calculate this value? This is where differentiation and integration come together in one of the most important theorems in all of mathematics.

The Fundamental Theorem of Calculus provides a powerful link between derivatives and integrals. It tells us how to solve a definite integral without having to sum up an infinite number of tiny rectangles (which is the formal definition of an integral).

The theorem states that if $F(x)$ is an antiderivative of $f(x)$, then:

This is a stunning result. To find the area under the curve $f(x)$ from $a$ to $b$, you simply need to:

1. Find the antiderivative, $F(x)$.
2. Evaluate the antiderivative at the upper limit, $F(b)$.
3. Evaluate the antiderivative at the lower limit, $F(a)$.
4. Subtract the two results.

Notice that the constant of integration $C$ disappears in this process, since $(F(b) + C) - (F(a) + C) = F(b) - F(a)$.

This theorem is the bedrock of calculus. It confirms that differentiation and integration are inverse operations, two sides of the same coin. One measures the rate of change, and the other measures the total accumulation of that change.

You now have the foundational tools to understand what integrals are and how they relate to the derivatives you've already mastered.