4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
255
(d)
1
−1
x
5 dx
(e)
2
0
(3x
3 − 2x
2 − e
x ) dx
⊳
Basic antiderivatives
The general problem of finding an antiderivative is difficult. In part, this is due to the
fact that we are trying to undo the process of differentiating, and the undoing is much
more difficult than the doing. For example, while it is evident that an antiderivative of
f (x) = sin(x) is F(x) = − cos(x) and that an antiderivative of g(x) = x 2 is G(x) =
1
3 x 3 ,
combinations of f and g can be far more complicated. Consider such functions as
5 sin(x) − 4x
2
, x
2 sin(x),
sin(x)
x 2 , and sin(x
2 ).
What is involved in trying to find an antiderivative for each? From our experience
with derivative rules, we know that while derivatives of sums and constant multiples
of basic functions are simple to execute, derivatives involving products, quotients, and
composites of familiar functions are much more complicated. Thus, it stands to reason
that antidifferentiating products, quotients, and composites of basic functions may be even
more challenging. We defer our study of all but the most elementary antiderivatives to
later in the text.
We do note that each time we have a function for which we know its derivative, we
have a function-derivative pair, which also leads us to knowing the antiderivative of a
function. For instance, since we know that
d
dx
[− cos(x)] = sin(x),
it follows that F(x) = − cos(x) is an antiderivative of f (x) = sin(x). It is equivalent to
say that f (x) = sin(x) is the derivative of F(x) = − cos(x), and thus F and f together
form the function-derivative pair. Clearly, every basic derivative rule leads us to such a
pair, and thus to a known antiderivative. In Activity 4.11, we will construct a list of most
of the basic antiderivatives we know at this time. Furthermore, those rules will enable
us to antidifferentiate sums and constant multiples of basic functions. For example, if
f (x) = 5 sin(x) − 4x 2 , note that since − cos(x) is an antiderivative of sin(x) and
1
3 x 3 is an
antiderivative of x 2 , it follows that
F(x) = −5 cos(x) −
4
3
x
3
is an antiderivative of f , by the sum and constant multiple rules for differentiation.
255
(d)
1
−1
x
5 dx
(e)
2
0
(3x
3 − 2x
2 − e
x ) dx
⊳
Basic antiderivatives
The general problem of finding an antiderivative is difficult. In part, this is due to the
fact that we are trying to undo the process of differentiating, and the undoing is much
more difficult than the doing. For example, while it is evident that an antiderivative of
f (x) = sin(x) is F(x) = − cos(x) and that an antiderivative of g(x) = x 2 is G(x) =
1
3 x 3 ,
combinations of f and g can be far more complicated. Consider such functions as
5 sin(x) − 4x
2
, x
2 sin(x),
sin(x)
x 2 , and sin(x
2 ).
What is involved in trying to find an antiderivative for each? From our experience
with derivative rules, we know that while derivatives of sums and constant multiples
of basic functions are simple to execute, derivatives involving products, quotients, and
composites of familiar functions are much more complicated. Thus, it stands to reason
that antidifferentiating products, quotients, and composites of basic functions may be even
more challenging. We defer our study of all but the most elementary antiderivatives to
later in the text.
We do note that each time we have a function for which we know its derivative, we
have a function-derivative pair, which also leads us to knowing the antiderivative of a
function. For instance, since we know that
d
dx
[− cos(x)] = sin(x),
it follows that F(x) = − cos(x) is an antiderivative of f (x) = sin(x). It is equivalent to
say that f (x) = sin(x) is the derivative of F(x) = − cos(x), and thus F and f together
form the function-derivative pair. Clearly, every basic derivative rule leads us to such a
pair, and thus to a known antiderivative. In Activity 4.11, we will construct a list of most
of the basic antiderivatives we know at this time. Furthermore, those rules will enable
us to antidifferentiate sums and constant multiples of basic functions. For example, if
f (x) = 5 sin(x) − 4x 2 , note that since − cos(x) is an antiderivative of sin(x) and
1
3 x 3 is an
antiderivative of x 2 , it follows that
F(x) = −5 cos(x) −
4
3
x
3
is an antiderivative of f , by the sum and constant multiple rules for differentiation.