Antiderivatives and Indefinite Integration

When we differentiate a constant (like 5 or -100), it becomes zero. This creates a problem: many different functions can have the same derivative. For example, $x^2 + 10$ and $x^2 - 50$ both have the derivative $2x$. To account for this 'lost' information, we must always add the constant of integration, denoted as $+C$. This $+C$ represents an infinite family of curves that are all vertical shifts of one another. Without it, your solution is incomplete because you've ignored all the other possible starting points.

Just as we memorized derivative rules for $e^x$, $\sin(x)$, and $\cos(x)$, we must learn their inverses. Since the derivative of $\sin(x)$ is $\cos(x)$, the integral $\int \cos(x) dx = \sin(x) + C$. However, be careful with signs: since the derivative of $\cos(x)$ is $-\sin(x)$, the integral $\int \sin(x) dx = -\cos(x) + C$. For exponentials, the rule is the most straightforward: $\int e^x dx = e^x + C$. These rules are the 'building blocks' for solving complex rate-of-change equations in physics and engineering.

While the general solution includes $+C$, we can find a particular solution if we are given an 'initial condition'—a specific point $(x, y)$ that the original function must pass through. This is called an Initial Value Problem. By plugging the known $x$ and $y$ values into our general antiderivative, we can solve for the specific value of $C$. This is how scientists determine the exact position of a rocket if they know its starting point and its velocity function.