ABSTRACT
If F(x) is differentiable for all values of x in the interval (a, b) and satisfies the equation dy dx f x/ = ( ), then F(x) is an integral of f(x) with respect to x. The notation is F x f x dx( ) ( )= ∫ or, in differential form, dF x f x dx( ) ( ) .=
For any function F(x) that is an integral of f(x) it follows that F(x) + C is also an integral. We thus write
∫ = +f x dx F x C( ) ( ) . (See Table of Integrals)
Let f(x) be defined on the interval [a, b], which is partitioned by points x x x xj n1 2 1, , , , ,… … − between a = x0 and b = xn. The jth interval has length Δx x xj j j= − −1, which may vary with j. The sum Σ Δjn j jf v x=1 ( ) , where vj is arbitrarily chosen in the jth subinterval, depends on the numbers x xn0 , , and the choice of the v as well as f, but if such sums approach a common value as all ∆x approach zero, then this value is the definite integral of f over the interval (a, b) and is denoted
f x dx a
b ( ) .≡ The fundamental theorem of integral calcu-
lus states that
f x dx F b F a a
( ) ( ) ( ),= −∫ where F is any continuous indefinite integral of f in the interval (a, b).
