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* = *x*^{4}, then an antiderivative of *f* is
*F* = *x*^{5}, which can be found by reversing the power rule.
Notice that not only is *x*^{5} an antiderivative of *f*, but
so are *x*^{5} + 4, *x*^{5} + 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.

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:

*x*^{n}*dx*=*x*^{n+1}+*c*as long as*n*does not equal -1. This is essentially the power rule for derivatives in reverse*cf*(*x*)*dx*=*c**f*(*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*

sec^{2}(*x*)*dx*= tan(*x*) +*c*

These are the opposite of the trigonometric derivatives.

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