MTH1170 Antiderivatives

Definition In calculus, an antiderivative of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F ′ = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative.

The Goal Using the symbolic representation F ‘ = f, we want to be able to find F given f.

Example Given the following functions and their derivatives, specify the antiderivatives of 2x, and 3x^2.

Constants & Antiderivatives As we continue to find antiderivatives it becomes obvious that more than one function differentiates to the same derivative. This is because the derivative of a constant is equal to zero.

Example Find the derivative of the following functions:

Constant of Integration To account for this we add a constant C to the antiderivatives that we find. We call this the constant of integration. The general antiderivative of f(x) = 2x is then F(x) = x^2 + C. Where the constant of integration C can be any real number.

The Anti-Power Rule Solve the following derivative:

The Anti-Power Rule Here we can see that the general antiderivative of x^n will be: