Page 103 - Schaum's Outline of Theory and Problems of Advanced Calculus

94 INTEGRALS [CHAP. 5

CONNECTING INTEGRAL AND DIFFERENTIAL CALCULUS
In the late seventeenth century the key relationship between the derivative and the integral was
established. The connection which is embodied in the fundamental theorem of calculus was responsible
for the creation of a whole new branch of mathematics called analysis.

Definition: Any function F such that F ðxÞ¼ f ðxÞ is called an antiderivative, primitive,or indefinite
0
integral of f .
The antiderivative of a function is not unique. This is clear from the observation that for any
constant c
0 0
ðFðxÞþ cÞ ¼ F ðxÞ¼ f ðxÞ
The following theorem is an even stronger statement.

Theorem. Any two primitives (i.e., antiderivatives), F and G of f differ at most by a constant, i.e.,
FðxÞ GðxÞ¼ C.

(See the problem set for the proof of this theorem.)
ð x 3
2
2
2
EXAMPLE. If F ðxÞ¼ x ,then FðxÞ¼ x dx ¼ þ c is an indefinite integral (antiderivative or primitive) of x .
0
3
The indefinite integral (which is a function) may be expressed as a definite integral by writing
x
ð ð
f ðtÞ dt
f ðxÞ dx ¼
c
The functional character is expressed through the upper limit of the definite integral which appears
on the right-hand side of the equation.
This notation also emphasizes that the definite integral of a given function only depends on the limits
of integration, and thus any symbol may be used as the variable of integration. For this reason, that
variable is often called a dummy variable. The indefinite integral notation on the left depends on
continuity of f on a domain that is not described. One can visualize the definite integral on the
right by thinking of the dummy variable t as ranging over a subinterval ½c; xŠ. (There is nothing unique
about the letter t; any other convenient letter may represent the dummy variable.)
The previous terminology and explanation set the stage for the fundamental theorem. It is stated in
two parts. The first states that the antiderivative of f is a new function, the integrand of which is the
derivative of that function. Part two demonstrates how that primitive function (antiderivative) enables
us to evaluate definite integrals.

THE FUNDAMENTAL THEOREM OF THE CALCULUS
Part 1 Let f be integrable on a closed interval ½a; bŠ. Let c satisfy the condition a @ c @ b, and
define a new function
ð x
f ðtÞ dt if a @ x @ b
FðxÞ¼
c
Then the derivative F ðxÞ exists at each point x in the open interval ða; bÞ, where f is continuous and
0
F ðxÞ¼ f ðxÞ. (See Problem 5.10 for proof of this theorem.)
0
Part 2 As in Part 1, assume that f is integrable on the closed interval ½a; bŠ and continuous in the
open interval ða; bÞ. Let F be any antiderivative so that F ðxÞ¼ f ðxÞ for each x in ða; bÞ.If a < c < b,
0
then for any x in ða; bÞ
ð x
f ðtÞ dt ¼ FðxÞ FðcÞ
c