Introduction to Integrals

Antiderivatives

An antiderivative of a function f is a function whose derivative is f. In other words, F is an antiderivative of f if F' = f. To find an antiderivative for a function f, we can often reverse the process of differentiation.

For example, if f = x⁴, then an antiderivative of f is F =

x⁵, which can be found by reversing the power rule. Notice that not only is

x⁵ an antiderivative of f, but so are

x⁵ + 4,

x⁵ + 6, etc. In fact, adding or subtracting any constant would be acceptable.

This should make sense algebraically, since the process of taking the derivative (i.e. going from F to f) eliminates the constant term of F.

Because a single continuous function has infinitely many antiderivatives, we do not refer to "the antiderivative", but rather, a "family" of antiderivatives, each of which differs by a constant. So, if F is an antiderivative of f, then G = F + c is also an antiderivative of f, and F and G are in the same family of antiderivatives.

Indefinite Integral

The notation used to refer to antiderivatives is the indefinite integral.

f (x)dx means the antiderivative of f with respect to x. If F is an antiderivative of f, we can write

f (x)dx = F + c. In this context, c is called the constant of integration.

To find antiderivatives of basic functions, the following rules can be used:

xⁿdx = xⁿ⁺¹ + c as long as n does not equal -1. This is essentially the power rule for derivatives in reverse
cf (x)dx = cf (x)dx. That is, a scalar can be pulled out of the integral.
(f (x) + g(x))dx = f (x)dx + g(x)dx. The antiderivative of a sum is the sum of the antiderivatives.
sin(x)dx = - cos(x) + c
cos(x)dx = sin(x) + c
sec²(x)dx = tan(x) + c
These are the opposite of the trigonometric derivatives.