TITLE: Fundamental Theorem of Calculus: Why Doesn't the Integral Depend on Lower Bound?
QUESTION [10 upvotes]: The second part of the Fundamental Theorem of Calculus essentially states that if $$F(x) = \int^x_a{f(t)}\,dt\,,$$ then $$F'(x) = f(x)\,.$$ My question is: why does the result not depend on the lower limit of integration $a$?

REPLY [23 votes]: Let, e.g., $b<a$. Then 
$$
G(x) := \int^x_b{f(t)}\,dt=\int_b^a {f(t)}\,dt+\int_a^x {f(t)}\,dt=F(x)+D,
$$
and the derivatives of $F$ and $G$ are identical.