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.
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