Diferentes versiones de Teorema Fundamental delCalculo

Juan Carlos Ponce Campuzanoponcecampuzanocarlos@gmail.com

4 de mayo de 2013

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Indice

1. Demostraciones donde se usa el teorema del valor medio para integra-les 4

1.1. Courant, R. & Fritz, J. (1965). Introduction to Calculus and Analysis.Interscience Publishers. USA. pp. 184-188. . . . . . . . . . . . . . . . . 4

1.2. Khuri, A. I. (2003). Advance Calculus with Applications in Statistics.John Wiley & Sons, Inc. New Jersey. pp. 218-219. . . . . . . . . . . . . 5

1.3. Loomis, L. H. & Sternberg, S. (1990). Advanced Calculus. Jones andBartlett Publishers. USA. p. 238. . . . . . . . . . . . . . . . . . . . . . 6

1.4. Apostol, T. M. (1967). Calculus, Volume 1, One-Variable Calculus withan Introduction to Linear Algebra, 2nd Edition.Waltham, MA: Blaisdell.pp. 247-250. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5. Olmsted, J. H. (1961). Advanced Calculus. Prentice Hall, Inc. USA. pp.128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Demostraciones donde se usa el teorema del valor medio 12

2.1. Marsden, J. & Weinstein, A. (1981). Calculus Unlimited. The BenjaminCummings Publishing Company, Inc. USA. p. 171. . . . . . . . . . . . 12

2.2. Bartle, R. G. (1927). Introduction to real analysis. Courier Companies,Inc. USA. pp. 251-253. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3. Rudin, W. (1981). Principios de analisis matematico. Mexico. MacGraw-Hill. p. 144. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4. Widder, D. V. (1989). Advanced Calculus. Dover Publications, Inc. NewYork. p. 150. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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2.5. Franklin, P. (1940). A Treatise on Advanced Calculus. John Wiley &Sons, Inc. USA. pp. 201-207 . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6. Spivak, M.(1994). Calculus. (Third Edition). Publish or Perish, Inc.,Houston. pp. 399-406. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3. Demostracion usando estimacion de errores 24

3.1. Edwards, H. M. (1980). Advanced Calculus. Robert E. Krieger PublishingCompany, Inc. USA. p. 52 . . . . . . . . . . . . . . . . . . . . . . . . . 24

4. Otra aproximacion usando la teorıa de sucesiones de funciones poli-gonales 27

4.1. Kuratowski, K. (1962). Introduction to Calculus. Addison Wesley Publis-hing Company, Inc. Massachusetts. . . . . . . . . . . . . . . . . . . . . 27

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1. Demostraciones donde se usa el teorema del valor

medio para integrales

1.1. Courant, R. & Fritz, J. (1965). Introduction to Calculusand Analysis. Interscience Publishers. USA. pp. 184-188.

The integral, the Primitive Function, and the Fundamental Theorems ofthe Calculus

As already stated, the connection between integration and differentiation is thecornerstone of the diferential and integral calculus.

We recall from section 2.4 that an indefinite integral of a continuous function f(x)is defined as a function φ(x) of the upper en point of integration by the formula

φ(x) =

∫ x

α

f(u)du,

where α was any point in the domain of f . We shall now prove.

FUNDAMENTAL THEOREM OF CALCULUS (Part one). The indefinite integralφ(x) of a continuous function f(x) always possesses a derivative φ′(x), and moreover

φ′(x) = f(x).

That is, differentiation of the indefinite integral of a continuous function alwaysreproduces the integrand

d

dx

∫ x

α

f(u)du = f(x).

This inverse character of the operations of differentiation and integration is the basicfact of calculus. The proof is an immediate consequence of the mean value theorem of

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integral calculus. According to that theorem we have for any values x and x + h of thedomain of f

φ(x + h)− φ(x) =

∫ x+h

x

f(u)du = hf(ξ),

where ξ is some value in the interval with end points x and x + h.

For h tending to zero the value ξ must tend to x so that

lımh→0

φ(x + h)− φ(x)

h= lım

h→0f(ξ) = f(x),

since f is continuous. Hence φ′(x) = f(x) as stated by theorem.

FUNDAMENTAL THEOREM OF CALCULUS. Every primitive function F (x) ofa given function f(x) continuous on an interval can be represented in the form

F (x) = c + φ(x) = c +

∫ x

a

f(u)du,

where c and a are constants, and conversely, for any constant values of c and a chosenarbitrarily this expression always represents a primitive function.

1.2. Khuri, A. I. (2003). Advance Calculus with Applicationsin Statistics. John Wiley & Sons, Inc. New Jersey. pp.218-219.

Theorem 6.4.8. Suppose that f(x) is continuous on [a, b]. Let F (x) =∫ x

af(t)dt.

Then we have the following:

i. dF (x)/dx = f(x), a ≤ x ≤ b.

ii.∫ b

af(x)dx = G(b)−G(a), where G(x) = F (x)+ c, and c is an arbitrary constant.

Proof. We have

dF (x)

dx=

d

dx

∫ x

a

f(t)dt = lımh→0

1

h

[∫ x+h

a

f(t)dt−∫ x

a

f(t)dt

]5

= lımh→0

1

h

∫ x+h

x

f(t)dt, by Theorem 6.4.4

= lımh→0

f(x + θh),

by Theorem 6.4.6, where 0 ≤ θ ≤ 1. Hence,

dF (x)

dx= lım

h→0f(x + θh) = f(x)

by the continuity of f(x). This result indicates that an indefinite integral of f(x) isany function whose derivative is equal to f(x). It is therefore unique up to a constant.Thus both F (x) and F (x) + C, where c is an arbitrary constant, are considered to beindefinite integrals.

To prove the second part of the theorem, let G(x) be defined on [a, b] as

G(x) = F (x) + c =

∫ x

a

f(t)dt + c

that is, G(x) is an indefinite integral of f(x) If x = a, then G(a) = c, since F (a) = 0.

Also, if x = b, then G(b) = F (b) + c =∫ b

af(t)dt + G(a). It follows that∫ b

a

f(x)dx = G(b)−G(a).

1.3. Loomis, L. H. & Sternberg, S. (1990). Advanced Calculus.Jones and Bartlett Publishers. USA. p. 238.

Theorem 10.3. If f ∈ C ([a, b] , W ) and F : [a, b] → W is defined by F (x) =∫ x

af(t)dt, then F ′ exists on (a, b) and F ′(x) = f(x).

Proof. By the continuity of f at x0, for every ε there exists a δ such that

‖f(x0)− f(x)‖ < ε

whenever |x− x0| < δ. But then∥∥∥∥∫ x

x0

(f(x0)− f(t)) dt

∥∥∥∥ ≤ ε |x− x0|

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and since∫ x

x0f(x0)dt = f(x0)(x−x0) by the definition of the integral for an elementary

function, we see that ∥∥∥∥f(x0)−(∫ x

x0

f(t)/(x− x0)

)dt

∥∥∥∥ < ε

Since∫ x

x0f(x0)dt = F (x) − F (x0), this is exactly the statement that the difference

quotient for F converges to f(x0), as was o be proved.

1.4. Apostol, T. M. (1967). Calculus, Volume 1, One-VariableCalculus with an Introduction to Linear Algebra, 2ndEdition.Waltham, MA: Blaisdell. pp. 247-250.

Teorema 1 Teorema del Valor Medio para Integrales. Si f es continua en [a, b], paraun cierto c de [a, b] tenemos ∫ b

a

f(x)dx = f(c)(b− a).

Teorema 2 Primera parte del Teorema Fundamental del Calculo. Sea f una funcionintegrable en [a, x] para todo x ∈ [a, b]. Sea c tal quea ≤ c ≤ b y A(x) =

∫ x

cf(t)dt si

a ≤ x ≤ b. Existe entonces A′(x) en cada punto x del intervalo (a, b) en el que f escontinua, y para tal x tenemos

A′(x) = f(x). (1)

Primer prueba:

Sea f una funcion continua en [x, x + h], con h positivo, entonces tenemos que∫ x+h

x

f(t)dt =

∫ x+h

c

f(t)dt−∫ x

c

f(t)dt = A(x + h)− A(x).

Por el teorema del valor medio para integrales, tenemos

A(x + h)− A(x) = hf(z),

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donde x ≤ z ≤ x + h. Luego, resulta

A(x + h)− A(x)

h= f(z),

y, puesto que x ≤ z ≤ x + h, encontramos que f(z) → f(x) cuando h → 0 con valorespositivos. Si h → 0 con valores negativos, se razona en forma parecida. Por consiguiente,A′(x) existe y es igual a f(x).

Demostracion analıtica:

Sea x un punto en el que f es continua y supuesta x fija, se forma el cociente:

A(x + h)− A(x)

h.

Para demostrar el teorema se ha de probar que este cociente tiende a f(x) cuandoh → 0. El numerador es:

A(x + h)− A(x) =

∫ x+h

c

f(t)dt−∫ x

c

f(t)dt =

∫ x+h

x

f(t)dt.

Si en la ultima integral se escribe f(t) = f(x) + [f(t)− f(x)] resulta:

A(x + h)− A(x) =

∫ x+h

x

f(x)dt +

∫ x+h

x

[f(t)− f(x)]dt

= hf(x) +

∫ x+h

x

[f(t)− f(x)]dt,

de donde

A(x + h)− A(x)

h= f(x) +

1

h

∫ x+h

x

[f(t)− f(x)]dt. (2)

Por tanto, para completar la demostracion de la ecuacion (1), es necesario demostrarque

lımh→0

1

h

∫ x+h

x

[f(t)− f(x)]dt = 0.

En esta parte de la demostracion es donde se hace uso de la continuidad de f en x.

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Si se designa por G(h) el ultimo termino del segundo miembro de 2, se trata dedemostrar que G(h) → 0 cuando h → 0. Aplicando la definicion de lımite, se ha deprobar que para cada ε > 0 existe un δ > 0 tal que

G(h) < ε (3)

siempre que 0 < h < δ. En virtud de la continuidad de f en x, dado un ε existe unnumero positivo δ tal que

|f(t)− f(x)| < 1

2ε (4)

siempre que

x− δ < t < x + δ. (5)

Si se elige h de manera que 0 < h < δ, entonces cada t en el intervalo [x, x+h] satisface(5) y por tanto (4) se verifica para cada t de este intervalo. Aplicando la propiedad∣∣∣∫ x+h

xg(t)dt

∣∣∣ ≤ ∫ x+h

x|g(t)| dt, cuando g(t) = f(t) − f(x), de la desigualdad en (4) se

pasa a la relacion:∣∣∣∣∫ x+h

x

[f(t)− f(x)]dt

∣∣∣∣ ≤ ∫ x+h

x

|f(t)− f(x)| dt ≤∫ x+h

x

1

2ε dt =

1

2hε < hε.

Dividiendo por h se ve que (3) se verifica para 0 < h < δ. Si h < 0, un razonamientoanalogo demuestra que (3) se verifica siempre que 0 < |h| < δ, lo que completa lademostracion.

Teorema 3 Segunda parte del Teorema Fundamental del Calculo. Supongamos f con-tinua en un intervalo abierto I, y sea P una primitiva cualquiera de f en I. Entonces,para cada c y cada x en I, tenemos

P (x) = P (c) +

∫ x

c

f(t)dt.

Demostracion:

Pongamos A(x) =∫ x

cf(t)dt. Puesto que f es continua en cada x de I, el primer

teorema fundamental nos dice que A′(x) = f(x) para todo x de I. Es decir, A es una

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primitiva de f en I. Puesto que dos primitivas de f pueden diferir tan solo en unaconstante, debe ser A(x) − P (x) = k para una ciera constante k. Cuando x = c, estaformula implica −P (c) = k, ya que A(c) = 0. Por consiguiente, A(x)− P (x) = −P (c),de lo que obtenemos

P (x) = P (c) +

∫ x

c

f(t)dt.

1.5. Olmsted, J. H. (1961). Advanced Calculus. Prentice Hall,Inc. USA. pp. 128

Theorem I. Let f(x) be defined and continuous on a closed interval [a, b] or [b, a],and define the function F (x) on this interval:

F (x) =

∫ x

a

f(t)dt.

Then F (x) is differentiable there with derivative f(x):

F ′(x) = f(x).

If a primitive or antiderivative or indefinite integal of a given function isdefined to be any function whose derivative is the given function, Theorem I assertsthat any continuous funciton has a primitive. The next theorem gives a method forevaluating the definite integral of any continuous function in terms of a given primitive.

Theorem II. Fundamental Theorem of Integral Calculus. If f(x) is conti-nuous on the closed interval [a, b] and if F (x) is any primitive of f(x) on this interval,then ∫ b

a

f(x)dx = F (b)− F (a).

Proof. By Theorem I and the hypotheses of Theorem II, the two functions∫ x

af(t)dt

and F (x) have the same derivative on the given interval. Therefore, by Theorem II,306, they differ by a constant:∫ x

a

f(t)dt− F (x) = C. (6)

10

Substitution of x = a gives the value of this constant: C = −F (z). Upon substitutionfor C in (6) we have

∫ x

af(t)dt = F (x)− F (a) which, for the particular value x = b, is

the desired result.

By the virtue of the Fundamental Theorem of Integral Calculus, the integral symbol∫, suggested by the letter S (the definite integral is the limit of a sum), is appropriate for

the indefinite integral as well as the definite integral. If F (x) is an arbitrary indefiniteintegral of a function of a function f(x), we write the equation∫

f(x)dx = F (x) + C,

where C is an arbitrary constant of integration, and call the symbol∫

f(x)dx theindefinite integral of f(x). The Fundamental Theorem can thus be considered as anexpression of the relation between the two kinds of integrals, definite and indefinite.

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2. Demostraciones donde se usa el teorema del valor

medio

2.1. Marsden, J. & Weinstein, A. (1981). Calculus Unlimi-ted. The Benjamin Cummings Publishing Company, Inc.USA. p. 171.

Statement of the Fundamental Theorem

Theorem 1 Fundamental Theorem of Calculus: Suppose that be function Fis differentiable everywhere on [a, b] and that F ′ is integrable on [a, b]. Then∫ b

a

F ′(x)dx = F (b)− F (a).

We will now give a complete proof of the fundamental theorem of calculus. Thebasic idea is as follows: Letting F be an antiderivative for f on [a, b], we will show thatif Lf and Uf are any lower and upper sums for f on [a, b], then Lf ≤ F (b)−F (a) ≤ Uf .Since f is assumed to be integrable on [a, b], the only number which can separate the

lower sums from the upper sums in this way is the integral∫ b

af(t)dt. It will follow that

F (b)− F (a) must equal∫ b

af(t)dt.

To show that every lower sum is less than or equal to F (b) − F (a), we must takeany piecewise constant g on [a, b] such that g(t) ≤ f(t) for all t in (a, b) and show that∫ b

ag(t)dt ≤ F (b) − F (a). Let (t0, t1, . . . , tn) be a partition adapted to g and let ki be

tha value of g on (ti−1, ti). Since F ′ = f , we have

ki = g(t) ≤ f(t) = F ′(t)

Henceki ≤ F ′(t)

for all t in (ti−1, ti). It follows from Corollary 1 of the mean value theorem (see pag.93-94) that

ki ≤F (ti)− F (ti−1)

ti − ti−1

12

Henceki∆ti ≤ F (ti)− F (ti−1)

Summing from i = 1 to n, we get

n∑i=1

ki∆ti ≤n∑

i=1

[F (ti)− F (ti−1)]

The left-hand side is just∫ b

ag(t)dt, by the definition of the integral of a step function.

The right-hand side is a telescopic sum equal to F (n)− F (t0). Thus we have∫ b

a

g(t)dt ≤ F (b)− F (a)

which is what we wanted to prove.

In the same way, we can show that if h(t) is a piecewise constant function such thatf(t) ≤ h(t) for all t in (a, b), then

F (b)− F (a) ≤∫ b

a

h(t)dt

as required. This completes the proof of the fundamental theorem.

2.2. Bartle, R. G. (1927). Introduction to real analysis. Cou-rier Companies, Inc. USA. pp. 251-253.

7.3.1 Fundamental Theorem of Calculus (First Form) Let f : [a, b] → R beintegrable on [a, b] and let F : [a, b] → R satisfy the conditions:

(a) F is continuous on [a, b];

(b) the derivative F ′ exists and F ′(x) = f(x) for all x ∈ (a, b).

Then: ∫ b

a

f(x)dx = F (b)− F (a) (7)

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Proof. Let ε > 0 be given; by the Riemann Criterion 7.1.8, there is a partitionP = (x0, x1, . . . , xn) of [a, b] such that

U(P ; F ′)− L(P ; F ′) < ε.

If we now apply the Mean Value Theorem 6.2.4 to F on each of the intervals [xk−1, xk],we obtain a point tk ∈ (xk−1, xk), such that

F (xk)− F (xk−1) = (xk − xk−1)F′(tk),

whence it follows that

m′k(xk − xk−1) ≤ F (xk)− F (xk−1) ≤ M ′

k(xk − xk−1),

where m′k and M ′

k denote the infimum and supremum of F ′ on [xk−1, xk]. If we addthese inequalities over all subintervals in the partition P and note that the middle term“telescopes”, we obtain

L(P ; F ′) ≤ F (b)− F (a) ≤ U(P ; F ′).

But since we also have

L(P ; F ′) ≤∫ b

a

F ′ ≤ U(P ; F ′),

it follows (why?) that ∣∣∣∣∫ b

a

F ′ − [F (b)− F (a)]

∣∣∣∣ < ε.

Since ε > 0 is aribitrary, equation (7) follows.

7.3.2 Corollary Let F : [a, b] → R satisfy the conditions:

(i) the derivative F ′ of F exists on [a, b],

(ii) the function F ′ is integrable on [a, b].

Then the equation (7) holds with f = F ′.

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7.3.3 Fundamental Theorem of Calculus (Second Form) Let f : [a, b] → Rbe integrable on [a, b] and let

F (x) =

∫ x

a

f for x ∈ [a, b]; (8)

then F is continuous on [a, b]. Moreover, if f is continuous at a point c ∈ [a, b], then Fis differentiable at c and

F ′(c) = f(c). (9)

Proof. Let K > 0 be such that |f(x)| ≤ K for x ∈ [a, b]. If x, y ∈ [a, b] and x < y,then since

F (y)− F (x) =

∫ y

a

f −∫ x

a

f =

∫ y

x

f

it follows from Corollary 7.2.6(a) that

|F (y)− F (x)| ≤ H |y − x| . (10)

The continuity of F follows form (10).

Now suppose that f is continuous at a point c ∈ [a, b]. Let ε > 0 be given and letδ > 0 be such that if |h| < δ and c + h ∈ [a, b], then |f(c + h)− f(c)| < ε. For any such

h we use the observation that (1/h)∫ c+h

c1dx = 1 to obtain∣∣∣∣f(c + h)− f(c)

h− f(c)

∣∣∣∣ =

∣∣∣∣1h∫ c+h

c

f(x)dx− f(c)

h

∫ c+h

c

1dx

∣∣∣∣=

1

|h|

∣∣∣∣∫ c+h

c

(f(x)− f(c)) dx

∣∣∣∣But since the integrand in the last integral is, in absolute value, less than ε, we inferfrom 7.26(a) that ∣∣∣∣f(c + h)− f(c)

h− f(c)

∣∣∣∣ ≤ 1

|h|ε |h| = ε.

15

Since ε > 0 is arbitrary, it follows that

lımh→0

F (c + h)− F (c)

h= f(c).

Hence we have F ′(c) = f(c).

7.3.4 Corollary Let f : [a, b] → R be continuous on [a, b] and let

F (x) =

∫ x

a

f for x ∈ [a, b].

Then F is differentiable on [a, b] and F ′(x) = f(x) for all [a, b].

Proof. This result follows immediately from the theorem.

It is sometimes useful to combine these two forms into one theorem, which wenow present. Note that the hypotheses of this version are stronger than in the earlierforms. However, the conclusion emphasizes the inverse nature of differentiation andintegrations for continuous functions.

7.3.5 Fundamental Theorem of Calculus (Combined form) Let F and f becontinuous on [a, b] and let F (a) = 0. Then the following statements are equivalent:

(i) F ′(x) = f(x) for all x ∈ [a, b];

(ii) F (x) =∫ x

af for all x ∈ [a, b].

Proof. The reader should check that the equivalence of these conditions is guaran-teed by Corollaries 7.3.2 and 7.3.4.

2.3. Rudin, W. (1981). Principios de analisis matematico.Mexico. MacGraw-Hill. p. 144.

6.21 El teorema fundamental del calculo. Si f ∈ R sobre [a, b] y si existe unafuncion diferenciable F sobre [a, b] tal que F ′ = f , entonces∫ b

a

f(x)dx = F (b)− F (a).

16

Demostracion Dado ε > 0, elıjase una particion P = {x0, . . . , xn} de [a, b], de talmanera que U(P, f)− L(P, f) < ε. El teorema del valor medio proporciona los puntosti ∈ [xi−1, xi] de tal manera que

F (xi)− F (xi−1) = f(ti)∆xi

para i = 1, . . . , n. Entonces

n∑i=1

f(ti)∆xi = F (b)− F (a).

Y del Teorema 6.7 (c) se deduce ahora que∣∣∣∣F (b)− F (a)−∫ b

a

f(x)dx

∣∣∣∣ < ε.

Por eso se verifica para cada ε > 0, la demostracion queda concluida.

2.4. Widder, D. V. (1989). Advanced Calculus. Dover Publi-cations, Inc. New York. p. 150.

Theorem A. f(x) ∈ C, a ≤ x ≤ b ⇒∫ b

af(x)dx exists.

Theorem B.

1. f(x) ∈ C, a ≤ x ≤ b

2. F ′(x) = f(x)

⇒∫ b

af(x)dx = F (b)− F (a).

This result is kwnon as the fundamental theorem of integral calculus. We include aproof for purposes of review. By Theorem A the limit (1) exists uniquely, no matterhow the points ξk are chosen. For our present convenience we choose them in a specialway, namely, so that

F (xk)− F (xk−1) = F ′(ξk)(xk − xk−1)

17

xk−1 < ξ < xk. This is possible by Theorem 2, Chapter 1. By hypothesis 2 the sum (1)becomes

n∑k=1

[F (xk)− F (xk−1)] = F (b)− F (a)

That is, it does nor change as n →∞, δ → 0, and hence has the stated limit.

2.5. Franklin, P. (1940). A Treatise on Advanced Calculus.John Wiley & Sons, Inc. USA. pp. 201-207

126. Evaluation of the Integral. Suppose that f(x) is a function integrable overthe interval a, b and throughout this interval is knwon to be the derivative of a secondfunction F (x), so that

f(x) = F ′(x).

Then we may apply the law of finite increments of section 74 to the function F (x)for each of the subintervals used in section 121, and so find points ξi such that

F (xi)− F (xi−1) = F ′(ξi)(xi − xi−1). (11)

Now use these values as the ξi in the sum for S in equation

S =n∑

i=1

f(ξi)δi.

We shall then have:

f(ξi)δi = F ′(ξi)(xi − xi−1) = F (xi)− F (xi−1),

so that:

S = f(ξ1)δ1 + f(ξ2)δ2 + · · ·+ f(ξn)δn

= F (x1)− F (x0) + F (x2)− F (x1) + · · ·+ F (xn)− F (xn−1)

= F (xn)− F (x0) = F (b)− F (a).

18

Thus we may form a particula sequence St for which the ξi are always chosen so thatequation (11) is satisfied, and for this sequence the values of St are all equal, so thatthe limit is F (b)− F (a).

Since f(x) is integrable, the limit will be the same for all other sequences, and hencewe shall have: ∫ b

a

f(x)dx = F (b)− F (a).

We write F (x)|ba in place of F (b)− F (a).

If f(x) is an integrable function over the interval c, d and F (x) is any indefiniteintegral of f(x), so that f(x) = F ′(x) throughout the closed interval c, d, then∫ b

a

f(x)dx = F (x)|ba = F (b)− F (a)

for a and b, any two points of the closed interval c, d.

129. Derivatives of integrals.

Let the function f(x) be integrable on the closed interval a, b and continuous at x0,a point of this interval. Then the integral of f(x) over the interval a, x is a function ofx, and we may write as in equation:

A(x) =

∫ x

a

f(x)dx =

∫ x

a

f(u)du,

where we have replaced the dummy variable of integration x by u so as to avoid confusionbetween the variable of integration and the variable upper limit. Let us now calculatethe derivative of A(x) at x0, using the fundamental definition. We have for any valueof x and x + h in the closed interval a, b:

A(x + h)− A(x) =

∫ x+h

a

f(u)du−∫ x

a

f(u)du =

∫ x+h

x

f(u)du.

by equation (29). We next deduce from the first mean value theorem for integrals thatfor a suitable intermediate value ξ, ξ between x and x + h, or ξ = x + θh, 0 < θ < 1,∫ x+h

x

f(u)du = hf(ξ) = hf(x + θh).

19

This shows thatA(x + h)− A(x)

h= hf(x + θh),

so that

lımh→0

A(x + h)− A(x)

h= f(x0),

since f(x) is continuous at x0. Thus the function A(x) has a derivative at x0, and

A′(x0) = f(x0)

We mar formulate the result as a theorem:

If f(x) is integrable in some closed interval a, b and if f(x) is continuous at somepoint x0 in the open interval a, b, then the function A(x) =

∫ x

af(x)dx or

∫ x

af(u)du has

a derivative for x = x0, and this derivative A′(x0) = f(x0).

2.6. Spivak, M.(1994). Calculus. (Third Edition). Publish orPerish, Inc., Houston. pp. 399-406.

TEOREMA 1 (PRIMER TEOREMA FUNDAMENTAL DEL CALCULO INFI-NITESIMAL)

Sea f integrable sobre [a, b] y defınase F sobre [a, b] por

F (x) =

∫ x

a

f.

Si f es continua en c de [a, b], entonces F es derivable en c, y

F ′(c) = f(c).

DEMOSTRACION

Supondremos que c esta en (a, b); el lector podra suplir las faciles modificacionesnecesarias para c = a o b. Por definicion

F ′(c) = lımh→0

F (c + h)− F (c)

h.

20

Supongamos primero que h > 0. Entonces

F (c + h)− F (c) =

∫ c+h

c

f.

Defimanos mk yMk como sigue:

mh = inf {f(x) : c ≤ x ≤ c + h} ,

Mh = sup {f(x) : c ≤ x ≤ c + h} .

Del teorema 13-7 se sigue que

mh · h ≤∫ c+h

c

f ≤ Mh · h.

Por lo tanto

mh ≤F (c + h)− F (c)

h≤ Mh.

Si h ≤ 0, solamente habra que cambiar unos pocos detalles del razonamiento. Sea

mh = inf {f(x) : c + h ≤ x ≤ c} ,

Mh = sup {f(x) : c + h ≤ x ≤ c} .

Entonces

mh · (−h) ≤∫ c

c+h

f ≤ Mh · (−h).

Por ser

F (c + h)− F (c) =

∫ c+h

c

f = −∫ c

c+h

f,

se obtiene

mh ≥F (c + h)− F (c)

h≥ Mh.

Puesto que h < 0, la division por h invierte de nuevo la desigualdad, obteniendose elmismo resultado que antes:

mh ≤F (c + h)− F (c)

h≤ Mh.

21

Esta igualdad se cumple para cualquier funcion integrable, sea o no continua. Sin em-bargo, puesto que f es continua en c,

lımh→0

mh = lımh→0

Mh = f(c),

y esto demuestra que

F ′(c) = lımh→0

F (c + h)− F (c)

h= f(c).

TEOREMA 2 (SEGUNDO TEOREMA FUNDAMENTAL DEL CALCULO IN-FINITESIMAL)

Si f es integrable sobre [a, b] y f = g′ para alguna funcion g, entonces∫ b

a

f = g(b)− g(a).

Sea P = {t0, t1, . . . , tn} una particion cualquiera de [a, b]. Segun el teorema del valormedio existe un punto xi en [ti−1, ti] tal que

g(ti)− g(ti−1) = g′(xi)(ti − ti−1)

= f(xi)(tt − ti−1).

Si

mi = inf {f(x) : ti−1 ≤ x ≤ ti} ,

Mi = sup {f(x) : ti−1 ≤ x ≤ ti} ,

entonces evidentemente

mh(ti − ti−1) ≤ f(xi)(tt − ti−1) ≤ Mh(ti − ti−1).

es decir,

mh(ti − ti−1) ≤ g(ti)− g(ti−1) ≤ Mh(ti − ti−1).

Sumando estas ecuaciones para i = 1, . . . , n obtenemos

n∑i=1

mh(ti − ti−1) ≤ g(b)− g(a) ≤n∑

i=1

Mh(ti − ti−1)

22

de manera que

L(f, P ) ≤ g(b)− g(a) ≤ U(f, P )

para toda particion P . Pero esto significa que

g(b)− g(a) =

∫ b

a

f.

23

3. Demostracion usando estimacion de errores

3.1. Edwards, H. M. (1980). Advanced Calculus. Robert E.Krieger Publishing Company, Inc. USA. p. 52

The Fundamental Theorem of Calculus

The evaluation of integrals in elemetary calculus is accomplished by the FundametalTheorem of Calculus, which can be stated as follows:

I. Let F (t) be a function for which the derivative F ′(t) exists and is a continuousfunction for t in the interval a ≤ t ≤ b. Then∫ b

a

f(x)dx = F (b)− F (a)

II. Let f(t) be a continuous function on a ≤ t ≤ b. Then there exists a differentiablefunctions F (t) on a ≤ t ≤ b such that f(t) = F ′(t).

Part I says that in order to evaluate a given integral it suffices to write the integrandas a derivative so that the desired integral is on the left side of equiation and a knownnumber is on the right.

Part II says that theoretically this procedure always works, that is, theoretically anycontinuous integrand can be written as a derivativ.

Proof of I

The idea of the theorem is the following: Let a = t0, < t1 < t2 . . . < tn = b be asubdivision of the interval into small intervals. Then

F (b)− F (a) = [F (b)− F (tn−1)] + [F (tn−1)− F (tn−2)] + . . . + [F (t1)− F (a)]

=∑

∆F =∑ ∆F

∆t∆t

where∑

denotes a sum over all intervals of the subdivision, and where for each subin-terval {ti−1 ≤ t ≤ ti} the symbol ∆F denots F (ti) − F (ti−1) and ∆t denotes ti − ti−1.

24

By the definition of ‘derivative’, the numbers ∆F∆t

are nearly F ′(t) for t in the interval;hence, by the definition of ‘integral’, the sum

∑∆F∆t

∆t is exactly equal to F (b)− F (a)this is the statement to be proved.

To make this rough argument into a proof of I, one must estimate the error ofaproximations ∫ ti

ti−1

F ′(t)dt ∼ F ′(t)∆t ∼ ∆F

∆t∆t.

In doing this is helpful to divide the difference between∫ ti

ti−1F ′(t)dt and ∆F by ∆t and

to estimate

1

∆t

{∫ ti

ti−1

F ′(t)dt−∆F

}, (12)

which can be thought of as the average difference per unit length between the numbers∫ titi−1

F ′(t)dt and ∆F = F (ti)−F (ti−1). Assuming that the theorem is true, this average

difference per unit length is of couser zero for all subintervals {ti−1 ≤ t ≤ ti} and this isthe statement to be proved. If the interval is further subdivided, then the maximum ofthis average, like any average, can only increase; that is, the average on at least one ofthe smaller intervals is as larger as the average over the whole interval. Since the limitof (12) as ∆t → 0 is F ′(t)−F ′(t) = 0 this observation will suffice to prove the theorem.

Specifically, for any r, s in the interval a ≤ r ≤ s ≤ b let εrs denote

εrs =1

s− r

{∫ s

r

F ′(t)dt− [F (s)− F (r)]

}.

If c in the midpoint between a and b then

εrs =1

b− a

{∫ b

a

F ′(t)dt− [F (b)− F (a)]

}=

1

b− a

{∫ c

a

F ′(t)dt +

∫ b

c

F ′(t)dt− [F (b)− F (c)]− [F (c)− F (a)]

}=

c− a

b− aεac +

b− c

b− aεcb

=1

2(εac + εcb) .

25

Thus either |εac| ≥ |εcb| or |εcb| ≥ |εab|; that is, the average error is at least as great on(at least) one of the two halves as it is on the whole interval. Dividing this half intohalves and repeating the argument shows that there is a quarter of the original intervalon which |ε| is at least |εab|. Continuing this process ad infinitum gives a sequence ofintervals such that the ith interval is one of the halves of the (i− 1)st (the first intervalis {a ≤ t ≤ b}) and such that the average error εi per unit length on the ith intervalsatisfies |εi| ≥ |εi−1|. As i → ∞ the intervals shrink down to a point,* say T , and εi

approaches

lım∆t→0

1

∆t

∫F ′(t)dt− lım

∆t→0

∆F

∆t= F ′(T )− F ′(T ) = 0

by (v) of §2.6 and by the definition of the derivative F ′(T ). Thus |εi| ≥ |εab| for all iand lımi→∞ εi = 0, which implies εab = 0. This completes the proof if I.

Proof of II

Given a continuous function f(t) on a ≤ t ≤ b, the integrand defines a function

F (c) =

∫ c

a

f(t)dt (13)

assigning numbers (the integral) to points c in the interval a ≤ c ≤ b. It is to be shownthat the function F so defined is differentiable and that its derivarite is f . But since

F (t1)− F (t0)

t1 − t0=

1

t1 − t0

∫ t1

t0

f(t)dt,

this follow immediately from (v) of §2.6.

26

4. Otra aproximacion usando la teorıa de sucesiones

de funciones poligonales

4.1. Kuratowski, K. (1962). Introduction to Calculus. Addi-son Wesley Publishing Company, Inc. Massachusetts.

9.1 Definition

A function F is called a primitive function of a function f defined in an open interval(finite or infinite), if F ′(x) = f(x) for every x.

For example the function sin x is a primitive function of the function cos x. Anyfunction of the form sin x+C, where C is a constant, is also a primitive function of thefunction cos x.

If a function f is defined in a closed interval a ≤ x ≤ b, then the function F is calledits primitive function, if F ′(x) = f(x) for a ≤ x ≤ b, F ′

+(a) = f(a) and F ′−(b) = f(b).

1.If two functions F and G are primitive functions of a function f in an interval ab(open or closed), then these two functions differ by a constant.

Indeed, if F ′(x) = G′(x), then, according to Theorem 4 of 7.5, there exists a constantC such that G(x) = F (x) + C for every x.

Conversely, a function obtained by adding a constant to a primitive function of afunction f is also a primitive function of the function f . Thus the expression F (x) + Cis the general form of a primitive function of the function f . We indicate this expressionby the symbol

∫f(x)dx (“the integral f(x)dx”) and we call it the indefinite integral of

the function f . So we have

(1)∫

f(x)dx = F (x) + C, where F ′(x) = f(x),

(2) ddx

∫f(x)dx = f(x),

(3)∫ dF (x)

dx= F (x) + C.

27

The evaluation of the indefinite integral of a function f , i.e. the calculation of theprimitive function of a function f is called the integration of the function f . So inte-gration is an inverse operation to differentiation. It follows from the definition of theindefinite integral, that any formula for the derivative of a function automatically givesa formula for the integral of another function (namely, the derived function). For exam-ple from the formula sin x

dx= cos x, we obtain

∫cos xdx = sin x+C. In general, however,

the problem of the calculation of the integral of a continuous function, which we do notknow to be a derivative of a certain function, is more difficult than the problem of diffe-rentiation. As we have seen in 7, the differentiation of functions which are compositionsof elementary functions does not lead out of their domain; yet an analogous theorem forindefinite integrals would not be true. It is known only that every continuous functionpossesses an indefinite integral (cf. 9.2). However, this theorem gives us no practicalprocedure of evaluating the indefinite integral of a given continuous function.

Remark. If the domain of x for which the equation F ′(x) = f(x) is satisfied is notan interval (finite or infinite), then it cannot be stated that the expression F (x) + Cgives all primitive functions of the function f in this domain of arguments.

For example, log |x|+C gives all primitive functions of the function 1x

in each of thetwo domains x < 0 and x > 0, separately, but not in the whole domain of x real anddifferent from 0. Namely, the function G(x) defined as log |x| for x < 0 and as log |x|+1for x 6= 0 is a primitive function of the function 1

xfor all x 6= 0 although it is not given

by the formula log x + C.

Let us complete now Theorem 1 as follows:

2. Let a point x0 be given inside an interval ab and let an arbitrary real number y0 begiven. If a function f possesses a primitive function in the interval ab, then it possessesone and only one primitive function F such that F (x0) = y0

Let P (x) be an arbitrary primitive function of the function f(x) in an interval ab.Let us write F (x) = P (x) − P (x0) + y0. Then we have F ′(x) = P ′(x) = f(x) andF (x0) = y0. Hence the function F satisfies the conditions of the theorem.

Moreover, it is the only function satisfying these conditions, since any other primitivefunction of the function f is of form F (x)+C, where C 6= 0, and this implies F (x0)+C =y0 + C 6= y0.

28

Geometrically, this theorem means that given any point on the plane with the abs-cissa belonging to the interval ab, there exists an integral curve (i.e. the graph of aprimitive function) passing through this point. The integral curves being parallel oneto another, only one integral curve of a given function f may pass through a given pointon the plane.

9.2. The integral of the limit. Integrability of continuous functions

We have proved in 7.9 that given a sequence of functions Fn(x), a < x < b, such thatthe derivatives are continuous and uniformly convergent in the interval ab to a functiong(x) and that the sequence Fn(c) is convergent for a certain point c belonging to theinterval ab, the sequence Fn(x) is convergent for every x belonging to this interval;moreover, writing F (x) = lımn=∞ Fn(x), we have F ′(x) = g(x).

Hence, the following lemma follows:

1. If the functions fn(x) are continuous and uniformly convergent in an intervalab to a function f(x) and if they possess primitive functions, then the function f(x)possesses also a primitive function.

The primitive functions Fn(x) may be chosen in such a way (according to 9.1, 2)that the equation Fn(c) = 0 holds for a certain point c of the interval ab, for each valueof n. Then ∫

f(x)dx = lımn=∞

Fn(x) + C,

i.e. the integral of the limit equals to the limit of the integrals.

Indeed, to obtain our lemma it is sufficient to substitute in the above formulationfn in place of F ′

n and f in place of g.

2. THEOREM. Every function continuous in an interval ab possesses a primitivefunction in this interval.

By the theorem proved in 6.4, every function continuous in the interval ab is alimit of a uniformly convergent sequence of polygonal functions. Hence, according tothe lemma it remains only to prove that any polygonal function f possesses a primitivefunction.

29

According to the definition of a polygonal function there exists a system of n +1 points a0 < al < . . . < an where a0 = a, an = b, and two systems of numbersc1, c2, . . . , cn andd1, d2, . . . , dn such that f(x) = ckx + dk for ak−1 < x < ak (k =1, 2, . . . , n).

Let us write

Fk(x) =1

2ckx

2 + dkx + ek for ak−1 ≤ x ≤ ak

where e1 = 0 and

ek+1 =1

2cka

2k + dkak + ek − (

1

2ck+1a

2k + dk+1ak)

for k ≥ 1.

Then Fk(ak) = Fk+1(ak), whence the collection of the functions F1, F2, . . . , Fn definesone function equal to each of the functions of this collection in the suitable interval,respectively.

Differentiating the function Fk we obtain immediately F ′(x) = f(x), i.e. the functionF is primitive with respect to the function f .

30

Referencias

[1] Apostol, T. M. (1967). Calculus, Volume 1, One-Variable Calculus with an Intro-duction to Linear Algebra, 2nd Edition.Waltham, MA: Blaisdell.

[2] Bartle, R. G. (1927). Introduction to real analysis. Courier Companies, Inc. USA.

[3] Courant, R. (1968). Differential and integral calculus. New York: Interscience Pu-blishers.

[4] Edwards, H. M. (1980). Advanced Calculus. Robert E. Krieger Publishing Com-pany, Inc. USA.

[5] Franklin, P. (1940). A Treatise on Advanced Calculus. John Wiley & Sons, Inc.USA.

[6] Khuri, A. I. (2003). Advance Calculus with Applications in Statistics. John Wiley& Sons, Inc. New Jersey.

[7] Kuratowski, K. (1962). Introduction to Calculus. Addison-Wesley Publishing Com-pany, Inc. Massachusetts.

[8] Loomis, L. H. & Sternberg, S. (1990). Advanced Calculus. Jones and Bartlett Pu-blishers. USA.

[9] Marsden, J. & Weinstein, A. (1981). Calculus Unlimited. The Benjamin CummingsPublishing Company, Inc. USA.

[10] Olmsted, J. H. (1961). Advanced Calculus. Prentice Hall, Inc. USA.

[11] Rudin, W. (1981). Principios de analisis matematico. (Lic. Miguel Iran AlcerrecaSanchez Trad.). Mexico. MacGraw-Hill.

[12] Spivak, M.(1994). Calculus. (Third Edition). Publish or Perish, Inc., Houston.

[13] Widder, D. V. (1989). Advanced Calculus. Dover Publications, Inc. New York.

31