As the name implies, the Fundamental
Theorem of Calculus (FTC) is among the biggest ideas
of calculus, tying together derivatives and integrals. The
first part of the theorem (FTC 1)
relates the rate at which an integral is growing to the function
being integrated, indicating that integration and differentiation
can be thought of as inverse operations. The second part of
the theorem (FTC 2) gives us an
easy way to compute definite integrals.
The Fundamental Theorem of Calculus
relates three very different concepts:
The definite integral
$\int_a^b f(x)\, dx$ is the limit of a sum. $$\int_a^b f(x) \,dx
= \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x,$$ where
$\Delta x = (b-a)/n$ and $x_i^*$ is an arbitrary point somewhere
between $x_{i-1}=a + (i-1)\Delta x$ and $x_i = a + i \Delta x$.
The name we give to the variable of integration doesn't matter:
$$\int_a^b f(x) \,dx = \int_a^b f(s)\, ds = \int_a^b f(t)\, dt$$
The integral function is
the function $$I(x) = \int_a^x f(s)\, ds.$$ That is, it is a
running total of the amount of stuff that $f$ represents,
between $a$ and $x$. If $f$ is the height of a curve, then
$I(x)$ is the area under the curve between $a$ and $x$. If
$f$ is velocity, then $I(x)$ is the distance traveled between
time $a$ and time $x$.
An antiderivative $F(x)$
of $f(x)$ is a function with $F'(x)=f(x)$. There are actually
many different anti-derivatives of $f(x)$, but they differ by
constants. For instance, $x^3$ and $x^3+7$ are both
anti-derivatives of $3x^2$.
When studying the Fundamental Theorem of
Calculus, it's very important to keep these
straight. This video explains the three different concepts.