This section discusses the concept of antiderivative and the theorms related to it.



A derivative provides an easy way to find the rate of changes for a function. On the other hand, an antiderivative is a way to find the function of a given derivative.

An antiderivative of a function f is any function F such that F' = f. It is a reverse process of differentiation.

    Indefinite Integral


The notation used to refer to antiderivative is the indefinite integral. ∫f(x)dx means the antiderivative of f with respect to x . If F(x) is an antiderivative of f(x), then

     ∫f(x)dx = F + c

in which c is an arbitrary constant.

Function Antiderivative
∫cf(x)dx cF(x) + c
∫xndx   (n ≠ -1) xn+1/(n + 1) + c
∫(f(x) + g(x))dx F(x) + G(x) + c
∫cosxdx sinx + c
∫sinxdx -cosx + c
∫sec2xdx tanx + c
∫secxtanxdx secx + c
∫cscxcotxdx -cscx + c
∫csc2xdx -cotx + c

The table on the left gives various examples of indefinite integrals.

The method to calculate the antiderivative or indefinite integral can be better understand by finding the antiderivative of function f '(x) = x2 when f(1) = 7/3.

The indefinite integral rules gives that


The antiderivative of the given function f '(x) = x2 can be expressed as


In order to determine the constant c, substituting f(1) = 7/3 into the above expression.

     f(1) = 1/3 + c = 7/3


Family of Curves for f(x) = x3/3 + c

Rearranging the above equation gives

     c = 2

Therefore, the solution of this example is

     f(x) = x3/3 + 2

Notice that the f(x) = x3/3 + c is a family of functions and a specific function depends of the given intial condition or boundary condition. Its graphic is shown on the left.