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4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
Finally, before proceeding to build a list of common functions whose antiderivatives
we know, we revisit the fact that each function has more than one antiderivative. Because
the derivative of any constant is zero, any time we seek an arbitrary antiderivative, we
may add a constant of our choice. For instance, if we want to determine an antiderivative
of g(x) = x 2 , we know that G(x) =
1
3 x 3 is one such function. But we could alternately
have chosen G(x) =
1
3 x 3 + 7, since in this case as well, G ′ (x) = x 2 . In some contexts later
on in calculus, it is important to discuss the most general antiderivative of a function. If
g(x) = x 2 , we say that the general antiderivative of g is
G(x) =
1
3
x
3 + C,
where C represents an arbitrary real number constant. Regardless of the formula for g,
including +C in the formula for its antiderivative G results in the most general possible
antiderivative.
Our primary current interest in antiderivatives is for use in evaluating definite integrals
by the Fundamental Theorem of Calculus. In that situation, the arbitrary constant C is
irrelevant, and thus we usually omit it. To see why, consider the definite integral
1
0
x
2 dx.
For the integrand g(x) = x 2 , suppose we find and use the general antiderivative G(x) =
1
3 x 3 + C. Then, by the FTC,
1
0
x
2 dx =
1
3
x
3 + C
1
0
=
1
3
(1)
3 + C
−
1
3
(0)
3 + C
=
1
3
+ C − 0 − C
=
1
3
.
Specifically, we observe that the C-values appear as opposites in the evaluation of the
integral and thus do not affect the definite integral’s value. In the same way, the potential
inclusion of +C with the antiderivative has no bearing on any definite integral, and thus
we generally choose to omit this possible constant whenever we evaluate an integral using
the Fundamental Theorem of Calculus.
In the following activity, we work to build a list of basic functions whose antiderivatives
we already know.
4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
Finally, before proceeding to build a list of common functions whose antiderivatives
we know, we revisit the fact that each function has more than one antiderivative. Because
the derivative of any constant is zero, any time we seek an arbitrary antiderivative, we
may add a constant of our choice. For instance, if we want to determine an antiderivative
of g(x) = x 2 , we know that G(x) =
1
3 x 3 is one such function. But we could alternately
have chosen G(x) =
1
3 x 3 + 7, since in this case as well, G ′ (x) = x 2 . In some contexts later
on in calculus, it is important to discuss the most general antiderivative of a function. If
g(x) = x 2 , we say that the general antiderivative of g is
G(x) =
1
3
x
3 + C,
where C represents an arbitrary real number constant. Regardless of the formula for g,
including +C in the formula for its antiderivative G results in the most general possible
antiderivative.
Our primary current interest in antiderivatives is for use in evaluating definite integrals
by the Fundamental Theorem of Calculus. In that situation, the arbitrary constant C is
irrelevant, and thus we usually omit it. To see why, consider the definite integral
1
0
x
2 dx.
For the integrand g(x) = x 2 , suppose we find and use the general antiderivative G(x) =
1
3 x 3 + C. Then, by the FTC,
1
0
x
2 dx =
1
3
x
3 + C
1
0
=
1
3
(1)
3 + C
−
1
3
(0)
3 + C
=
1
3
+ C − 0 − C
=
1
3
.
Specifically, we observe that the C-values appear as opposites in the evaluation of the
integral and thus do not affect the definite integral’s value. In the same way, the potential
inclusion of +C with the antiderivative has no bearing on any definite integral, and thus
we generally choose to omit this possible constant whenever we evaluate an integral using
the Fundamental Theorem of Calculus.
In the following activity, we work to build a list of basic functions whose antiderivatives
we already know.