Fundamental Theorems of Calculus

The first fundamental theorem of calculus states that, if is Continuous on the Closed Interval and is the Antiderivative (Indefinite Integral) of on , then

 (1)

The second fundamental theorem of calculus lets be Continuous on an Open Interval and lets be any point in . If is defined by

 (2)

then
 (3)

at each point in .

The complex fundamental theorem of calculus states that if has a Continuous Antiderivative in a region containing a parameterized curve for , then

 (4)