Understanding the connection between definite integrals and the Fundamental Theorem of Calculus (FTC) can be tough for many students. The FTC has two main parts that show how differentiation and integration work together, but figuring out what they really mean can be tricky.

1. Definite Integrals:

   • A definite integral, written as $\int_a^b f(x) \, dx$, finds the area under the curve of the function $f(x)$ between the points $x = a$ and $x = b$.
   • But for many students, the idea of area and how it connects to the function's behavior can be confusing.

2. Fundamental Theorem of Calculus:

   • The first part of the FTC says that if $F$ is an antiderivative of $f$, then you can calculate the definite integral like this: $\int_a^b f(x) \, dx = F(b) - F(a)$. This means you can use antiderivatives to find the definite integral, but figuring out the right $F$ isn't always easy.
   • Many students also find the symbols and ideas in both differentiation and integration hard to understand.

Even with these challenges, practice really helps! Breaking down the steps of integration can make a big difference. Looking at graphs and focusing on specific properties of integrals can also help make things clearer. With time, the relationship becomes easier to understand.