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 $a$?
Calculus – Why Integral Doesn’t Depend on Lower Bound
calculusintegration
Best Answer
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.