To use the Fundamental Theorem of Calculus effectively, you need to understand how differentiation and integration work together. This connection is really important for solving many calculus problems. Here’s a simple guide to help you through it.

First, get to know the theorem. It says that if $F$ is an antiderivative of a continuous function $f$ on the interval from $a$ to $b$, then:

$\int_a^b f(x) \, dx = F(b) - F(a)$

This means you can easily switch between finding an integral and using antiderivatives to evaluate it.

Next, let's identify the function you want to integrate. Check if your function is continuous over the range you care about. If it’s not, you might have to use different methods to solve your problem.

Once you have a good function, find its antiderivative. This process may include techniques like substitution or integration by parts. Make sure you can differentiate functions well because finding the antiderivative is like doing the reverse. Don’t forget about any constants of integration, but don’t worry too much about them, as they will cancel out when you calculate the definite integral.

After that, evaluate the antiderivative at the upper and lower limits of the interval. Substituting $b$ and $a$ into your antiderivative $F$ gives you $F(b) - F(a)$. This step is really important because it shows the area under the curve described by the function $f(x)$ from $a$ to $b$.

Finally, understand what the result means. The number you get tells you the total area between the function and the x-axis for that interval. Remember, areas above the x-axis add to the total, while areas below it subtract from it.

By following these steps—understanding the theorem, checking the function, finding the antiderivative, evaluating at the limits, and interpreting the result—you’ll be able to use the Fundamental Theorem of Calculus confidently.