# Fundamental Theorem of Integral Calculus

Theorem: Let f be continuous function defined on the closed interval [a, b] and F be an anti- derivative of f, then = [F(x)]ab = F(b) â€“ F(a), where a and b are called the limits of integration, a being a lower or inferior and b being the upper or superior limit.

Notes:
• If f(x) is not defined at x = a and x = band defined in open interval (ab) then  can be evaluated.
• If  then the equation f(x) = 0 has at least one root lying in (ab) provided f is a continuous function in (a,b).
• In , the function f needs to be well defined and continuous in [ab].

For instance, the consideration of definite integral  dx is erroneous since the function f expressed by f(x) =  is not defined in a portion â€“1 <x < 1 of the closed interval [â€“2, 3].