7.5 Definition of an Indefinite Integral [or primitive of f(x)] dx then a Suppose that f(x) is integrable in the interval  x x b d a (t )dt  f ( ) f x at each point x in the interval. A function F(x) with the property that F1(x) = f(x) at point x of the interval a  x b is called an indefinite integral or primitive of f(x) x ( )  F x Thus for a continuous function f(x) we have (t )dt f a b Consider if ( ) ( )  f x dx G x we say in G(x) is the a , b indefinite integral of ( ) f x dx a 7.6 The fundamental Theorem of the integral calculus If f(x) is continuous in b f(x) then ( ) (b ) (a )  f x dx G G a This is called the fundamental theorem of the integral calculus Proof Let F be given by 65 x ( )  F x (t f dt a Then a (a ) (t ) 0   F dt we have Since a ( ) 0 ( )    F a G a c ( ) ( )   F x G x c ( )  c a
7.5 Definition of an Indefinite Integral [or primitive of f(x)] Suppose that f(x) is integrable in the interval x d f (t )dt  f ( x ) at each point x in the interval. dx a a x b  then A function F(x) with the property that F1(x) = f(x) at point x of the interval a x b  is called an indefinite integral or primitive of f(x) Thus for a continuous function f(x) we have F ( x )  x f (t )dt a b Consider if a f ( x ) dx  G ( x ) , we say in G(x) is the indefinite integral of b f ( x ) dx a 7.6 The fundamental Theorem of the integral calculus If f(x) is continuous in f(x) then b f ( x ) dx  G (b ) G (a ) a This is called the fundamental theorem of the integral calculus Proof Let F be given by 65 F ( x )  x f (t dt a Then Since F ( x )  G ( x )  c a F (a )  (t ) dt  0 we have a F ( a )  0  G ( a )  c c  ( a )
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