The Difference and Connection between Antiderivative and Integral

The antiderivative is the inverse process of the derivative.The integral is to calculate the area closed by the graph of a function and x(or y) axis. However, Newton-Leibniz connect antiderivative and integration by giving a formula:∫^a_bf (x) dx = F(b)−F(a). In some cases, the integral of function f(x) on closed interval [a,b] can be calculated from the difference of its bounds’ antiderivative[(F(a)-F(b)]. When we do the calculation of antiderivative, we come up with a formula with an inconstant instead a definite value, which is the outcome of integral. A continues function must have its antiderivative. A function with first discontinuous points does not have its antiderivative.

Antiderivative and integral are two different things: if on interval[a,b], F’(x)=f(x), we can say F(x) is one of antiderivatives of f(x) on this interval. Based on this case, if F(x) is one of antiderivatives of f(x), F(x)+C(C is a random constant) is antiderivative of f(x). Then if f(x) is able to be integrated on [a,b], it may not have antiderivative. If f(x) is continuous on [a,b], f(x) constantly has its antiderivative G(x) and G(x) is its upper limit function: F(x)=∫^x_af(t)dt+C

In fact, the antiderivative equals indefinite integral. A continuous function can have infinite antiderivatives, but can have only 1 definite value of integral(interval must be given).