For g(x) = ∫ a to x f(t)dt, I would like to know why there is an x at the upper limit and why it not equal to b, what is the difference?
There's a very important difference- is a number, a constant. is a function of x.
For example if f(t)= 2t+ 1, then while .
More subtle is the difference between and the , the "indefinite integral" or "anti-derivative". where we need the constant C since for any C. The "a" in determines the specific C: while .