I want to understand the connection between the primitive function or antiderivative and the definite integral.
My problem with this is the independent variable called t in the formula for the first part of the Fundamental Theorem of Calculus.
Here's a composite of the answers I've already seen for this question. Because of t I don't understand it:
The primitive is a function $F(x)$ such that
$F′(x) = f(x)$
The first part of the FTC, which is the thing that connects differentiation/antidifferentiation to the definite integral, is this:
$$F(x) = \int_a^x f(t) dt $$ or “$F(x)$ is a primitive function of $f(x)$. The lower bounds a is fixed. The upper bounds $x$ is variable.”
You can write $\int f(x) dx$ as $$\int_a^x f(t) dt + C$$
A complete analysis of this problem is given in Richard Courant’s calculus book (linked below) page 109+.
Some of the sites I’ve seen: http://ia700700.us.archive.org/34/items/DifferentialIntegralCalculusVolI/Courant-DifferentialIntegralCalculusVolI.pdf
http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/ftc.html
web.utk.edu/~wneilson/mathbook.pdf page 153-154
www.physicsforums.com/showthread.php?t=212449&highlight=x+dillema
www.jirka.org/diffyqs/htmlver/diffyqsse3.html
math.stackexchange.com/questions/105937/what-does-integration-do
www.math.hmc.edu/calculus/tutorials/fundamental_thm/
www.intuitive-calculus.com/fundamental-theorem-of-calculus.html