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An antiderivative of a function

For example, if

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.

*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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