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.)
2. Definite Integral
Let f(x) be defined on the interval [a, b], which is partitioned by points x x x x
j n1 1, , , , ,2 … … − between a = x0 and b = xn. The jth interval has length ∆ x x xj j j= − −1 , which may vary with j. The sum Σ ∆
j j f v x=1 ( ) ,
where vj is arbitrarily chosen in the jth subinterval, depends on the numbers x x
n0 , ,… 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 ∫ a
b f x dx( ) . The fundamental theorem of integral calculus states that
b f x dx F b F a∫ = −( ) ( ) ( ),
where F is any continuous indefinite integral of f in the interval (a, b).
