# 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*)]*=*_{a}^{b}*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*=*b*and defined in open interval (*a*,*b*) then can be evaluated. - If then the equation
*f*(*x*) = 0 has at least one root lying in (*a*,*b*) provided*f*is a continuous function in (*a*,*b*). - In , the function
*f*needs to be well defined and continuous in [*a*,*b*].*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].