Chapter 4 Integration (積分)

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Chapter 4 Integration

(積分)

Hung-Yuan Fan (范洪源)

Department of Mathematics, National Taiwan Normal University, Taiwan

Fall 2018

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 4, Calculus B 1/119

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本章預定授課範圍

4.1 Antiderivatives and Indefinite Integration 4.2 Area

4.3 Riemann Sums and Definite Integrals 4.4 The Fundamental Theorem of Calculus 4.5 Integration by Substitution

4.6 Indeterminate Forms and L’Hôpital’s Rule 4.7 The Natural Logarithmic Function: Integration 4.8 Inverse Trigonometric Functions: Integration

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Section 4.1

Antiderivatives and Indefinite Integration

(反導函數與不定積分)

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Def (反導函數的定義)

A function F is an antiderivative (反導函數) of f on the interval I if F ^′ (x) = f(x) ∀ x ∈ I .

Example

The functions F ₁ (x) = x ³ , F ₂ (x) = x ³ − 5 and F 3 (x) = x ³ + 97 are antiderivatives of f(x) = 3x ² on R.

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Thm 4.1 (最廣反導函數的表示式)

If F is an antiderivative of f on the interval I, then

G is an antiderivative of f on I ⇐⇒ G(x) = F(x) + C, where C is a constant.

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Indefinite Integration (不定積分)

Def. of Indefinite Integrals

If F(x) is an antiderivative of f on the interval I, the indefinite integral of f w.r.t. x (函數 f 對 x 的不定積分) is defined by

Z

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

where f is called the integrand (被積函數) and C is called the constant of integration. (積分常數)

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Remark (積分與微分的關係) If F ^′ (x) = f(x) ∀ x ∈ I , then

Z

F ^′ (x) dx = Z

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

and d

dx

h Z f(x) dx i

= d dx

h F(x) + C i

= F ^′ (x) = f(x), i.e., the integration and differentiation are inverse operations to each other! (積分與微分互為反運算!)

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Thm (Basic Integration Rules; 1/4)

Suppose that the antiderivatives of f and g exist and let 0 ̸= k ∈ R.

(1) Z

k dx = kx + C.

(2) Z

[k · f(x)] dx = k · h Z

f(x) dx i .

(3) Z

[f(x) ± g(x)] dx = h Z

f(x) dx i

± h Z

g(x) dx i .

(4) Z

x ⁿ dx = 1

n + 1 x ⁿ⁺¹ + C for n ̸= −1 .

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Thm (Basic Integration Rules; 2/4) (5)

Z

cos x dx = sin x + C.

(6) Z

sin x dx = − cos x + C.

(7) Z

sec ² x dx = tan x + C.

(8) Z

sec x tan x dx = sec x + C.

(9) Z

csc ² x dx = − cot x + C.

(10) Z

csc x cot x dx = − csc x + C.

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Thm (Basic Integration Rules; 3/4) (11)

Z

e ^kx dx = 1

k e ^kx + C.

(12) Z

a ^kx dx = 1

k(ln a) a ^kx + C for 0 < a ̸= 1 . (13)

Z 1 x dx =

Z

x ⁻¹ dx = ln |x| + C.

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Thm (Basic Integration Rules; 4/4) (14)

Z 1

√ 1 − k ² x ² dx = 1

k sin ⁻¹ (kx) + C.

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Z 1

1 + k ² x ² dx = 1

k tan ⁻¹ (kx) + C.

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Z 1

x √

k ² x ² − 1 dx = sec ⁻¹ (kx) + C.

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Application of the Indefinite Integral

If g is conti. on the interval [x ₀ , ∞), consider the initial-value problem (I.V.P.; 初值問題)

( F ^′ (x) = g(x), (differential equation; D.E. 微分方程) F(x ₀ ) = y 0 . (initial condition; I.C. 初值條件) (1)

Main Questions

What is the general solution (通解) F(x) of the D.E.in (1)?

How to solve the particular solution (特解) of the I.V.P. (1)?

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Solutions of the I.V.P.

Integrating the D.E. in (1) w.r.t. x = ⇒ F(x) =

Z

F ^′ (x) dx = Z

g(x) dx = G(x) + C is the general solution of the D.E., where C is a constant.

Substituting the I.C. into the general solution = ⇒

y ₀ = F(x ₀ ) = G(x ₀ ) + C or C = y ₀ − G(x 0 ). So, the particular solution to I.V.P. (1) is given by

F(x) = F _p (x) = G(x) + y ₀ − G(x 0 ).

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Solutions of the I.V.P.

Integrating the D.E. in (1) w.r.t. x = ⇒ F(x) =

Z

F ^′ (x) dx = Z

g(x) dx = G(x) + C

is the general solution of the D.E., where C is a constant.

Substituting the I.C. into the general solution = ⇒

y ₀ = F(x ₀ ) = G(x ₀ ) + C or C = y ₀ − G(x 0 ). So, the particular solution to I.V.P. (1) is given by

F(x) = F _p (x) = G(x) + y 0 − G(x 0 ).

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Section 4.2 Area (面積)

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Def (Σ Notation)

The sum of real numbers a ₁ , a 2 , . . . , a n is denoted by X n

i=1

a _i = a 1 + a 2 + · · · + a n ,

where i is the index (足碼) of the summation and a _i is the ith term (第 i 項) of the sum.

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Thm 4.2 (Summation Formulas) (1) P ⁿ

i=1

c = c + c + · · · + c = cn for any constant c.

(2) P ⁿ

i=1

i = 1 + 2 + · · · + n = ⁿ⁽ⁿ⁺¹⁾ ₂ .

(3) P ⁿ

i=1

i ² = 1 ² + 2 ² + · · · + n ² = n(n+1)(2n+1)

6 .

(4) P ⁿ

i=1

i ³ = 1 ³ + 2 ³ + · · · + n ³ =

h n(n+1) 2

i 2

= ⁿ ² ⁽ⁿ⁺¹⁾ ₄ ² .

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Area of a Plane Region

Let f ≥ 0 be conti. on [a, b] , and consider the plane region defined by

R = {(x, y) ∈ R ² | a ≤ x ≤ b, 0 ≤ y ≤ f(x)}.

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Lower and Upper Sums (1/2)

Dividing [a, b] into n subintervals

[x 0 , x 1 ], · · · , [x i −1 , x i ], · · · , [x n −1 , x n ]

of equal width ∆x = ^b ^−a _n , where a = x ₀ < x ₁ < · · · < x n = b.

For each 1 ≤ i ≤ n, since f is conti. on each [x i −1 , x _i ], it follows from E.V.T. that ∃ m i , M i ∈ [x i −1 , x i ] s.t.

f(m _i ) ≤ f(x) ≤ f(M i ) ∀ x ∈ [x i −1 , x i ].

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Lower and Upper Sums (1/2)

Dividing [a, b] into n subintervals

[x 0 , x 1 ], · · · , [x i −1 , x i ], · · · , [x n −1 , x n ]

of equal width ∆x = ^b ^−a _n , where a = x ₀ < x ₁ < · · · < x n = b.

For each 1 ≤ i ≤ n, since f is conti. on each [x i −1 , x i ], it follows from E.V.T. that ∃ m i , M i ∈ [x i −1 , x i ] s.t.

f(m _i ) ≤ f(x) ≤ f(M i ) ∀ x ∈ [x i −1 , x i ].

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Lower and Upper Sums (2/2)

Def (下和與上和的定義)

(1) Lower sum (下和): s(n) = P ⁿ

i=1

f(m _i )∆x.

(2) Upper sum (上和): S(n) = P ⁿ

i=1

f(M _i )∆x.

Remark

From the definitions of s(n) and S(n), we see that s(n) ≤ area(R) ≤ S(n) ∀ n ∈ N.

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示意圖 (承上頁)

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Thm 4.3 (下和與上和的極限值) If f ≥ 0 is conti. on [a, b], then

n lim →∞ s(n) = A = lim

n →∞ S(n) ∃.

In this case, we know that area( R) = A ∃ by the Squeeze Thm.

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Def (非負連續函數與 x-軸所夾面積)

If f ≥ 0 is conti. on [a, b], then the area of the region R bounded by the graph of f, the x-axis, x = a and x = b is

A = area( R) = lim

n →∞

X n i=1

f(c _i )∆x,

where c _i ∈ [x i −1 , x i ] for i = 1, 2, . . . , n and ∆x = ^b ^−a _n .

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Section 4.3

Riemann Sums and Definite Integrals (黎曼和與定積分)

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Riemann Sums (黎曼和)

Def (黎曼和的定義)

Let f be defined on a closed interval I = [a, b].

(1) The set ∆ = {x 0 , x ₁ , · · · , x n −1 , x _n } is called a partition (分割) of I if a = x ₀ < x 1 < · · · < x n −1 < x n = b.

(2) The width of the ith subinterval [x _i ₋₁ , x i ] is ∆x i = x i − x i −1 for each i = 1, 2, . . . , n.

(3) The norm (範數) of a partition ∆ is defined by

∥∆∥ = max

1 ≤i≤n ∆x i .

(4) If c _i ∈ [x i −1 , x i ] for i = 1, 2, . . . , n, then P n i=1

f(c _i )∆x i is called a Riemann sum (黎曼和) of f for the partition ∆.

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Remarks

(1) For a general partition ∆ of [a, b], we see that b − a

n ≤ ∥∆∥ =⇒ b − a

∥∆∥ ≤ n.

So, ∥∆∥ → 0 ⇒ n → ∞ , but n → ∞ ; ∥∆∥ → 0 ! (2) If ∆x i = ^b ^−a _n ∀ i , then ∥∆∥ → 0 ⇐⇒ n → ∞ .

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Definite Integrals (定積分)

Def (定積分的定義)

Let f be defined on a closed interval I = [a, b].

(1) If the limit lim

∥∆∥→0

P n i=1

f(c _i )∆x i ∃ for any partition ∆ of I, then f is integrable (可積分的) on I. In this case, the limit

Z _b

a

f(x) dx = lim

∥∆∥→0

X n i=1

f(c _i )∆x i

is called the definite integral (定積分) of f from a to b.

(2) The number a is called the lower limit of integration (積分下限) and b is called the upper limit of integration (積分上限).

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Remarks (1)

Z

f(x) dx = F(x) + C denotes a family of functions, but Z b

a

f(x) dx is a real number.

(2) In general, the following notations Z _b

a

f(x) dx = Z _b

a

f(y) dy = Z _b

a

f(t) dt = Z _b

a

f(w) dw = · · · denote the same definite integral of f from a to b.

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Existence of Definite Integrals

Thm 4.4 (定積分的存在性)

If f is conti. on a closed interval I = [a, b], then f is integrable on I, i.e., the definite integral of f from a to b

Z b a

f(x) dx = lim

∥∆∥→0

X n i=1

f(c _i )∆x i ∃

for any partition ∆ of the interval I.

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General Version of Thm 4.4

f has at most finitely many discontinuities on [a, b] = ⇒ f is (Riemann) integrable on [a, b].

(函數 f 在 [a, b] 上最多僅有限個不連續點 = ⇒ f 在 [a, b] 上必定是一個黎曼可積的函數!)

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Thm 4.5 (非負連續函數與 x-軸所圍的區域面積) If f ≥ 0 is conti. on [a, b], then the area of the region

R = {(x, y) ∈ R ² | a ≤ x ≤ b, 0 ≤ y ≤ f(x)}

is given by A = area( R) = Z b

a

f(x) dx ≥ 0 .

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Def (Two Special Definite Integrals) (1)

Z _a

a

f(x) dx = 0 for any a ∈ R.

(2) If f is integrable on [a, b], then Z a

b

f(x) dx = − Z b

a

f(x) dx.

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Properties of Definite Integrals

Thm (定積分的性質)

Suppose that f and g are integrable on [a, b], and let k ∈ R.

(1) Z b

a

[k · f(x)] dx = k · Z ^b

a

f(x) dx .

(2) Z _b

a

[f(x) ± g(x)] dx = Z ^b

a

f(x) dx

± Z ^b

a

g(x) dx .

(3) Additivity (可加性):

Z _b

a

f(x) dx = Z _c

a

f(x) dx + Z

c

b f(x) dx for a < c < b.

(4) Preservation of Inequality (不等號的維持):

f(x) ≤ g(x) ∀ x ∈ [a, b] =⇒

Z b a

f(x) dx ≤ Z b

a

g(x) dx.

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示意圖 (承上頁)

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Note

In general, we know that Z _b

a

[f(x)g(x)] dx ̸= Z ^b

a

f(x) dx Z ^b

a

g(x) dx and Z b

a

f(x) g(x) dx ̸=

R b a f(x) dx R _b

a g(x) dx .

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Section 4.4

The Fundamental Theorem of Calculus (微積分基本定理; F.T.C.)

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Thm 4.9 (The First F.T.C.)

If f is conti. on I = [a, b] and F ^′ (x) = f(x) ∀ x ∈ I, then Z b

a

f(x) dx = F(b) − F(a) ≡ F(x) ^b

a ≡ F(x) i b a .

Note: the definite integral of f from a to b can be evaluated by the function values of antiderivative F(a) and F(b) directly!

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Example 2 of Section 4.3 (Revisited)

Applying the First F.T.C. (Thm 4.9) directly, it is easily seen that Z ₁

−2 2x dx = x ² ¹

−2 = 1 ² − (−2) ² = 1 − 4 = −3, since F(x) = x ² is an antiderivative of the integrand f(x) = 2x.

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Thm 4.10 (M.V.T. for Integrals; 積分的均值定理) If f is conti. on I = [a, b], then ∃ c ∈ [a, b] s.t.

f(c) = 1 b − a

Z b a

f(x) dx or f(c) · (b − a) = Z b

a

f(x) dx.

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Proof of Thm 4.10

Since f is conti. on I = [a, b], it follows that Z _b

a

f(x) dx ∃ and

∃ m, M ∈ I s.t. f(m) ≤ f(x) ≤ f(M) ∀ x ∈ I. We thus obtain

f(m)(b − a) = Z b

a

f(m) dx ≤ Z b

a

f(x) dx ≤ Z b

a

f(M) dx = f(M)(b − a)

or, equivalently, we see that f(m) ≤ 1

b − a Z _b

a

f(x) dx ≤ f(M).

Now, from I.V.T., ∃ c ∈ [a, b] s.t.

f(c) = 1 b − a

Z _b

a

f(x) dx ≡ f av .

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示意圖 (承上頁)

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Def (函數在閉區間上的平均值)

If f is conti. on I = [a, b], then the average value of f on I is given by f _av ≡ 1

b − a Z b

a

f(x) dx.

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The Definite Integral as a Function

If f is conti. on an open interval I containing a, define a real-valued function by

F(x) = Z _x

a

f(t) dt ∀ x ∈ I.

Some Questions

Is F differentiable on the open interval I?

If yes, what is the derivative of F?

Furthermore, is F an antiderivative of f on I?

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Thm 4.11 (The Second F.T.C.)

If f is conti. on an open interval I containing a, and define a real-valued function by

F(x) = Z _x

a

f(t) dt ∀ x ∈ I,

then F is diff. on I with the derivative F ^′ (x) = d

dx h Z ^x

a

f(t) dt i

= f(x) ∀ x ∈ I, i.e. F is an antiderivative of f on the open interval I.

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Proof of Thm 4.11

For any x ∈ I, we shall prove that

F ^′ (x) = lim

h →0

F(x + h) − F(x)

h = lim

h →0

∫ x+h

a f(t) dt − ∫ x a f(t) dt

h = f(x).

By M.V.T., ∃ c 1 ∈ [x, x + h] s.t. f(c ₁ ) = ¹ _h R x+h

x f(t) dt for h > 0, and ∃ c 2 ∈ [x + h, x] s.t. f(c ₂ ) = _−h ¹ R _x

x+h f(t) dt for h < 0. Thus,

h lim →0

⁺

F(x + h) − F(x)

h = lim

h →0

⁺

R x+h x f(t) dt

h = lim

h →0

⁺

f(c 1 ) = f(x) and

h lim →0

⁻

F(x + h) − F(x)

h = lim

h →0

⁻

R x

x+h f(t) dt

−h = lim

h →0

⁻

f(c 2 ) = f(x), since f is conti. on I. This completes the proof.

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Remark (Thm 4.11 的變形版本)

If f is conti. on an open interval I containing a, then it follows from Thm 4.11 that

d dx

h Z ^a

x

f(t) dt i

= d dx

h − Z _x

a

f(t) dt i

= −f(x) for all x ∈ I. (積分上下限顛倒會差了一個負號喔!)

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General Version of Thm 4.11

If f is conti. on an open interval I containing a, and u(x), v(x) are diff. functions of x with range(u), range(v) ⊆ I, then

(1) d dx

h Z ^u(x)

a

f(t) dt i

= f(u(x)) · u ^′ (x).

(2) d dx

h Z ^u(x)

v(x)

f(t) dt i

= f(u(x)) · u ^′ (x) − f(v(x)) · v ^′ (x).

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Section 4.5

Integration by Substitution (積分代換法)

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Main Goal

Applying a substitution u = g(x) to deal with the integral Z

f(g(x)) · g ^′ (x) dx = Z

f(u) du, and its associated definite integrals.

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Thm 4.13 (合成函數的反導函數)

Let g : D → I be a function with range(g) = I being an interval, and let f be conti. on I. If g is diff. on D and

F ^′ (x) = f(x) ∀ x ∈ I, then Z

f(g(x))g ^′ (x) dx = F(g(x)) + C, where C is a constant of integration.

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Change of Variables for Indefinite Integrals

Remark (The method of u-substitution; u-代換法)

If we let u = g(x), then du = g ^′ (x) dx and it follows from Thm 4.13 that

Z

f(u) du = Z

f(g(x))g ^′ (x) dx = F(g(x)) + C = F(u) + C, where C is a constant of integration.

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Thm 4.14 (General Power Rule for Integration) If u = g(x) is a diff. function of x and n ̸= −1 , then

Z

[g(x)] ⁿ g ^′ (x) dx = Z

u ⁿ du = u ⁿ⁺¹

n + 1 + C = [g(x)] ⁿ⁺¹ n + 1 + C,

where C is a constant of integration.

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Change of Variables for Definite Integrals

Thm 4.15 (定積分的 u-代換法)

If u = g(x) has a conti. derivative on I = [a, b] and f is conti. on range(g), then

Z b a

f(g(x))g ^′ (x) dx = Z g(b)

g(a)

f(u) du.

(將對 x 的定積分變數變換成對 u 的定積分!)

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Integration of Odd and Even Functions

Thm 4.16 (奇偶函數在 [ −a, a] 上的定積分)

Let f be integrable on the closed interval I = [ −a, a] with a > 0.

(1) If f is even, i.e., f( −x) = f(x) ∀ x ∈ I , then Z _a

−a f(x) dx = 2 Z _a

0 f(x) dx.

(2) If f is odd, i.e., f( −x) = −f(x) ∀ x ∈ I , then Z a

−a

f(x) dx = 0.

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示意圖 (承上頁)

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示意圖 (承上例)

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Section 4.6

Indeterminate Forms and L’Hôpital’s Rule

(不定型與羅必達法則)

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Types of Indeterminate Forms

For the limit of a quotient f(x)/g(x) as x → c, we may have

x lim →c

f(x) g(x) = 0

0 , ∞

∞ , 0 · ∞, 1 ^∞ , ∞ ⁰ , 0 ⁰ , ∞ − ∞, which are called the indeterminate forms (不定型).

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Thm 4.18 (L’Hôpital’s Rule; L’H Rule)

Assume that f and g are diff. on I = (a, c) ∪ (c, b) and g ^′ (x) ̸= 0 ∀ x ∈ I. If the limit

x lim →c

f(x) g(x) = 0

0 or ±∞

±∞ , then we have

x lim →c

f(x) g(x) = lim

x →c

f ^′ (x)

g ^′ (x) . (分子和分母各別微分喔!)

Note: Thm 4.18 also holds for the one-sided limits. (羅必達法則也適用於求解單邊極限值!)

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Remark (與 Thm 1.9 比較)

Thm 1.9 of Section 1.6 can be derived immediately from the L’Hôpital’s Rule, since we see that the limit is an indeterminate form of type 0 0 and hence

θ→0 lim sin θ

θ = lim

θ→0

cos θ

1 = cos(0) = 1.

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Indeterminate Forms of Type 0 · ∞ If lim

x →c f(x) = 0 and lim

x →c g(x) = ±∞, then consider

x lim →c [f(x)g(x)] = lim

x →c

f(x)

1/g(x) (Type 0 0 ) or

lim x →c [f(x)g(x)] = lim

x →c

g(x)

1/fx) (Type ∞

∞ ).

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Indeterminate Forms of Type 1 ^∞ , ∞ ⁰ or 0 ⁰ Assume that lim

x →c [f(x)] ^g(x) = 1 ^∞ , ∞ ⁰ , 0 ⁰ . If we know that lim x →c g(x) ln[f(x)] = L (Type 0

0 or ∞

∞ ) using the L’Hôpital’s Rule, then

lim x →c [f(x)] ^g(x) = lim

x →c e g(x) ln[f(x)] = e ^L .

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Example (補充題) Evaluate

x lim →∞ x ¹ ^x . (Type ∞ ⁰ ) Sol: From Example 2, we see that

x lim →∞

ln x x = lim

x →∞

1 x

1 = lim

x →∞

1 x = 0 by applying L’Hôpital’s Rule. So, we immediately obtain

x lim →∞ x ¹ ^x = lim

x →∞ e ^{ln x} ^x = e ⁰ = 1.

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Indeterminate Forms of Type ∞ − ∞

The original limit will become an indeterminate form of type 0 0 , if we apply the technique of reduction to common denominator.

(使用通分技巧將原極限問題變成標準不定型!)

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Example 7 (使用兩次羅必達法則) Evaluate the limit

lim

x →1 ⁺

1 ln x − 1 x − 1

. (Type ∞ − ∞)

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Solution of Example 7

Applying the L’Hôpital’s Rule twice, we see that lim

x →1 ⁺

1 ln x − 1 x − 1

= lim

x →1 ⁺

x − 1 − ln x

(x − 1) ln x (Type 0

0 ; 通分!)

= lim

x →1 ⁺

1 − 1/x

ln x + (x − 1)(1/x) (使用 L’H Rule!)

= lim

x →1 ⁺

1 − 1/x

ln x + (x − 1)(1/x) · x x

= lim

x →1 ⁺

x − 1

x ln x + x − 1 (Type 0 0 )

= lim

x →1 ⁺

1 ln x + x(1/x) + 1 (再次使用 L’H Rule!)

= 1

0 + 1 + 1 = 1

2 . (直接代入求極限喔!)

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Section 4.7

The Natural Logarithmic Function:

Integration

(自然對數函數的積分)

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Thm 4.19 (Log Rule for Integration) Let u = u(x) be a diff. function of x. Then

(1) Z 1

x dx = ln |x| + C, (2)

Z u ^′ (x) u(x) dx =

Z 1

u du = ln |u(x)| + C, where C is a constant of integration.

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Thm (三角函數的積分公式)

Let u = u(x) be a diff. function of x. Then (1)

Z

sin u du = − cos u + C, (2)

Z

cos u du = sin u + C, (3)

Z

tan u du = − ln | cos u| + C, (4)

Z

cot u du = ln | sin u| + C, (5)

Z

sec u du = ln | sec u + tan u| + C, (6)

Z

csc u du = − ln | csc u + cot u| + C, where C is a constant of integration.

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Chapter 4 Integration (積分)

Chapter 4 Integration

(積分)

Hung-Yuan Fan (范洪源)

Department of Mathematics, National Taiwan Normal University, Taiwan

Fall 2018

本章預定授課範圍

4.1 Antiderivatives and Indefinite Integration 4.2 Area

4.3 Riemann Sums and Definite Integrals 4.4 The Fundamental Theorem of Calculus 4.5 Integration by Substitution

4.6 Indeterminate Forms and L’Hôpital’s Rule 4.7 The Natural Logarithmic Function: Integration 4.8 Inverse Trigonometric Functions: Integration

Section 4.1

Antiderivatives and Indefinite Integration

(反導函數與不定積分)

Def (反導函數的定義)

A function F is an antiderivative (反導函數) of f on the interval I if F ′ (x) = f(x) ∀ x ∈ I .

Example

The functions F 1 (x) = x 3 , F 2 (x) = x 3 − 5 and F 3 (x) = x 3 + 97 are antiderivatives of f(x) = 3x 2 on R.

Thm 4.1 (最廣反導函數的表示式)

If F is an antiderivative of f on the interval I, then

G is an antiderivative of f on I ⇐⇒ G(x) = F(x) + C, where C is a constant.

Indefinite Integration (不定積分)

Def. of Indefinite Integrals

If F(x) is an antiderivative of f on the interval I, the indefinite integral of f w.r.t. x (函數 f 對 x 的不定積分) is defined by

Z

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

where f is called the integrand (被積函數) and C is called the constant of integration. (積分常數)

Remark (積分與微分的關係) If F ′ (x) = f(x) ∀ x ∈ I , then

Z

F ′ (x) dx = Z

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

and d

dx

h Z f(x) dx i

= d dx

h F(x) + C i

= F ′ (x) = f(x), i.e., the integration and differentiation are inverse operations to each other! (積分與微分互為反運算!)

Thm (Basic Integration Rules; 1/4)

Suppose that the antiderivatives of f and g exist and let 0 ̸= k ∈ R.

(1) Z

k dx = kx + C.

(2) Z

[k · f(x)] dx = k · h Z

f(x) dx i .

(3) Z

[f(x) ± g(x)] dx = h Z

f(x) dx i

± h Z

g(x) dx i .

(4) Z

x n dx = 1

n + 1 x n+1 + C for n ̸= −1 .

Thm (Basic Integration Rules; 2/4) (5)

Z

cos x dx = sin x + C.

(6) Z

sin x dx = − cos x + C.

(7) Z

sec 2 x dx = tan x + C.

(8) Z

sec x tan x dx = sec x + C.

(9) Z

csc 2 x dx = − cot x + C.

(10) Z

csc x cot x dx = − csc x + C.

Thm (Basic Integration Rules; 3/4) (11)

Z

e kx dx = 1

k e kx + C.

(12) Z

a kx dx = 1

k(ln a) a kx + C for 0 < a ̸= 1 . (13)

Z 1 x dx =

Z

x −1 dx = ln |x| + C.

Thm (Basic Integration Rules; 4/4) (14)

Z 1

√ 1 − k 2 x 2 dx = 1

k sin −1 (kx) + C.

(15)

Z 1

A function F is an antiderivative (反導函數) of f on the interval I if F ^′ (x) = f(x) ∀ x ∈ I .

The functions F ₁ (x) = x ³ , F ₂ (x) = x ³ − 5 and F 3 (x) = x ³ + 97 are antiderivatives of f(x) = 3x ² on R.

Remark (積分與微分的關係) If F ^′ (x) = f(x) ∀ x ∈ I , then

F ^′ (x) dx = Z

= F ^′ (x) = f(x), i.e., the integration and differentiation are inverse operations to each other! (積分與微分互為反運算!)

x ⁿ dx = 1

n + 1 x ⁿ⁺¹ + C for n ̸= −1 .

sec ² x dx = tan x + C.

csc ² x dx = − cot x + C.

e ^kx dx = 1

k e ^kx + C.

a ^kx dx = 1

k(ln a) a ^kx + C for 0 < a ̸= 1 . (13)

x ⁻¹ dx = ln |x| + C.

√ 1 − k ² x ² dx = 1

k sin ⁻¹ (kx) + C.

1 + k ² x ² dx = 1

k tan ⁻¹ (kx) + C.

k ² x ² − 1 dx = sec ⁻¹ (kx) + C.

If g is conti. on the interval [x ₀ , ∞), consider the initial-value problem (I.V.P.; 初值問題)

( F ^′ (x) = g(x), (differential equation; D.E. 微分方程) F(x ₀ ) = y 0 . (initial condition; I.C. 初值條件) (1)

F ^′ (x) dx = Z

y ₀ = F(x ₀ ) = G(x ₀ ) + C or C = y ₀ − G(x 0 ). So, the particular solution to I.V.P. (1) is given by

F(x) = F _p (x) = G(x) + y ₀ − G(x 0 ).

F ^′ (x) dx = Z

y ₀ = F(x ₀ ) = G(x ₀ ) + C or C = y ₀ − G(x 0 ). So, the particular solution to I.V.P. (1) is given by

F(x) = F _p (x) = G(x) + y 0 − G(x 0 ).

The sum of real numbers a ₁ , a 2 , . . . , a n is denoted by X n

a _i = a 1 + a 2 + · · · + a n ,

where i is the index (足碼) of the summation and a _i is the ith term (第 i 項) of the sum.

Thm 4.2 (Summation Formulas) (1) P ⁿ

(2) P ⁿ

i = 1 + 2 + · · · + n = ⁿ⁽ⁿ⁺¹⁾ ₂ .

(3) P ⁿ

i ² = 1 ² + 2 ² + · · · + n ² = n(n+1)(2n+1)

(4) P ⁿ

i ³ = 1 ³ + 2 ³ + · · · + n ³ =

= ⁿ ² ⁽ⁿ⁺¹⁾ ₄ ² .

R = {(x, y) ∈ R ² | a ≤ x ≤ b, 0 ≤ y ≤ f(x)}.

of equal width ∆x = ^b ^−a _n , where a = x ₀ < x ₁ < · · · < x n = b.

For each 1 ≤ i ≤ n, since f is conti. on each [x i −1 , x _i ], it follows from E.V.T. that ∃ m i , M i ∈ [x i −1 , x i ] s.t.

f(m _i ) ≤ f(x) ≤ f(M i ) ∀ x ∈ [x i −1 , x i ].

of equal width ∆x = ^b ^−a _n , where a = x ₀ < x ₁ < · · · < x n = b.

f(m _i ) ≤ f(x) ≤ f(M i ) ∀ x ∈ [x i −1 , x i ].

(1) Lower sum (下和): s(n) = P ⁿ