Properties of Definite Integrals Let f and g be integrable functions on [a, b] and let c a real number. Then 1 . edx = c (b-a) 2. c . f(x) d: = b 3. [( f (x ) + g(x)) dx = b f (x) dxt 9 ( x) dx a 4. " s(aldr = - Sf( x ) dx 5. / f(=) d: = O ( zero width integral )we can abbreviate this FTC Part 1 The Fundamental Theorem of Calculus, Part I. If f is continuous on a, b , then the function g defined by g (x ) = ( f(t)d, asisb is continuous on [a, b] and differentiable on (a, b) and g'(x) = f(x).The Fundamental Theorem of Calculus, Part II. If f is continuous on a, b], then f(x) dx = F(b) - F(a) where F is any antiderivative of f. That is, F is a function such that F = f.The Indefinite Integral . A function F is called an antiderivative of f on an interval / if F'(x) = f(x) for all r in I. . From the Fundamental Theorem of Calculus Part 1, we learned if f is continuous, then X fit ) at = F (x ) a is an antiderivative of f. . The most general form of the antiderivative of f is called the indefinite integral and is denoted [f ( x ) d x = F ( x ) + C where C is a real number