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# Advanced Calculus (I)

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(1)

## Advanced Calculus (I)

WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

(2)

## 5.4 Improper Riemann Integration

### Remark:

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

a

f (x )dx = lim

c→a+



d →b−lim Z d

c

f (x )dx

 .

WEN-CHINGLIEN Advanced Calculus (I)

(3)

## 5.4 Improper Riemann Integration

### Remark:

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

a

f (x )dx = lim

c→a+



d →b−lim Z d

c

f (x )dx

 .

(4)

### Proof:

By Theorem 5.26,

F (x ) = Z x

a

f (t)dt is continuous on [a,b]. Thus

Z b a

f (x )dx = F (b) − F (a)

= lim

c→a+( lim

d →b−F (d ) − F (c))

= lim

c→a+



d →b−lim Z d

c

f (x )dx

 .2

WEN-CHINGLIEN Advanced Calculus (I)

(5)

### Proof:

By Theorem 5.26,

F (x ) = Z x

a

f (t)dt is continuous on [a,b]. Thus

Z b a

f (x )dx = F (b) − F (a)

= lim

c→a+( lim

d →b−F (d ) − F (c))

= lim

c→a+

 lim

Z d

f (x )dx

 .2

(6)

### Proof:

By Theorem 5.26,

F (x ) = Z x

a

f (t)dt is continuous on [a,b]. Thus

Z b a

f (x )dx = F (b) − F (a)

= lim

c→a+( lim

d →b−F (d ) − F (c))

= lim

c→a+



d →b−lim Z d

c

f (x )dx

 .2

WEN-CHINGLIEN Advanced Calculus (I)

(7)

### Proof:

By Theorem 5.26,

F (x ) = Z x

a

f (t)dt is continuous on [a,b]. Thus

Z b a

f (x )dx = F (b) − F (a)

= lim

c→a+( lim

d →b−F (d ) − F (c))

= lim

c→a+

 lim

Z d

f (x )dx

 .2

(8)

### Proof:

By Theorem 5.26,

F (x ) = Z x

a

f (t)dt is continuous on [a,b]. Thus

Z b a

f (x )dx = F (b) − F (a)

= lim

c→a+( lim

d →b−F (d ) − F (c))

= lim

c→a+



d →b−lim Z d

c

f (x )dx

 .2

WEN-CHINGLIEN Advanced Calculus (I)

(9)

### Proof:

By Theorem 5.26,

F (x ) = Z x

a

f (t)dt is continuous on [a,b]. Thus

Z b a

f (x )dx = F (b) − F (a)

= lim

c→a+( lim

d →b−F (d ) − F (c))

= lim

c→a+

 lim

Z d

f (x )dx

 .2

(10)

Definition

Let [a,b] be a nonempty, open (possibly unbounded) interval and f : (a, b) →R.

(i)

f is said to be locally integrable on (a,b) if and only if f is integrable on each closed subinterval [c,d] of (a,b).

(ii)

f is said to be improperly integrable on (a,b) if and only if f is locally integrable on (a,b) and

(18)

Z b a

f (x )dx := lim

c→a+



d →b−lim Z d

c

f (x )dx



exists and is finite. This limit is called the improper (Riemann) integrable of f over (a,b).

WEN-CHINGLIEN Advanced Calculus (I)

(11)

Definition

Let [a,b] be a nonempty, open (possibly unbounded) interval and f : (a, b) →R.

(i)

f is said to be locally integrable on (a,b) if and only if f is integrable on each closed subinterval [c,d] of (a,b).

(ii)

f is said to be improperly integrable on (a,b) if and only if f is locally integrable on (a,b) and

(18)

Z b a

f (x )dx := lim

c→a+



d →b−lim Z d

c

f (x )dx



exists and is finite. This limit is called the improper

(12)

Definition

Let [a,b] be a nonempty, open (possibly unbounded) interval and f : (a, b) →R.

(i)

f is said to be locally integrable on (a,b) if and only if f is integrable on each closed subinterval [c,d] of (a,b).

(ii)

f is said to be improperly integrable on (a,b) if and only if f is locally integrable on (a,b) and

(18)

Z b a

f (x )dx := lim

c→a+



d →b−lim Z d

c

f (x )dx



exists and is finite. This limit is called the improper (Riemann) integrable of f over (a,b).

WEN-CHINGLIEN Advanced Calculus (I)

(13)

Definition

Let [a,b] be a nonempty, open (possibly unbounded) interval and f : (a, b) →R.

(i)

f is said to be locally integrable on (a,b) if and only if f is integrable on each closed subinterval [c,d] of (a,b).

(ii)

f is said to be improperly integrable on (a,b) if and only if f is locally integrable on (a,b) and

(18)

Z b a

f (x )dx := lim

c→a+



d →b−lim Z d

c

f (x )dx



exists and is finite. This limit is called the improper

(14)

Definition

Let [a,b] be a nonempty, open (possibly unbounded) interval and f : (a, b) →R.

(i)

f is said to be locally integrable on (a,b) if and only if f is integrable on each closed subinterval [c,d] of (a,b).

(ii)

f is said to be improperly integrable on (a,b) if and only if f is locally integrable on (a,b) and

(18)

Z b a

f (x )dx := lim

c→a+



d →b−lim Z d

c

f (x )dx



exists and is finite. This limit is called the improper (Riemann) integrable of f over (a,b).

WEN-CHINGLIEN Advanced Calculus (I)

(15)

Definition

Let [a,b] be a nonempty, open (possibly unbounded) interval and f : (a, b) →R.

(i)

f is said to be locally integrable on (a,b) if and only if f is integrable on each closed subinterval [c,d] of (a,b).

(ii)

f is said to be improperly integrable on (a,b) if and only if f is locally integrable on (a,b) and

(18)

Z b a

f (x )dx := lim

c→a+



d →b−lim Z d

c

f (x )dx



exists and is finite. This limit is called the improper

(16)

### Remark:

The order of the limits in (18) does not matter. In particular, if the limits in (18) exists, then

Z b a

f (x )dx = lim

d →b−



c→a+lim Z d

c

f (x )dx

 .

WEN-CHINGLIEN Advanced Calculus (I)

(17)

### Remark:

The order of the limits in (18) does not matter. In particular, if the limits in (18) exists, then

Z b a

f (x )dx = lim

d →b−



c→a+lim Z d

c

f (x )dx

 .

(18)

### Proof:

Let x0 ∈ (a, b) be fixed. By Theorem 5.20 and 3.8

c→a+lim



d →b−lim Z d

c

f (x )dx



= lim

c→a+

Z x0 c

f (x )dx + lim

d →b−

Z d x0

f (x )dx



= lim

c→a+

Z x0

c

f (x )dx + lim

d →b−

Z d x0

f (x )dx

= lim

d →b−



c→a+lim Z d

c

f (x )dx

 .2

WEN-CHINGLIEN Advanced Calculus (I)

(19)

### Proof:

Let x0 ∈ (a, b) be fixed. By Theorem 5.20 and 3.8

c→a+lim



d →b−lim Z d

c

f (x )dx



= lim

c→a+

Z x0 c

f (x )dx + lim

d →b−

Z d x0

f (x )dx



= lim

c→a+

Z x0

c

f (x )dx + lim

d →b−

Z d x0

f (x )dx

= lim

d →b−



c→a+lim Z d

c

f (x )dx

 .2

(20)

### Proof:

Let x0 ∈ (a, b) be fixed. By Theorem 5.20 and 3.8

c→a+lim



d →b−lim Z d

c

f (x )dx



= lim

c→a+

Z x0 c

f (x )dx + lim

d →b−

Z d x0

f (x )dx



= lim

c→a+

Z x0

c

f (x )dx + lim

d →b−

Z d x0

f (x )dx

= lim

d →b−



c→a+lim Z d

c

f (x )dx

 .2

WEN-CHINGLIEN Advanced Calculus (I)

(21)

### Proof:

Let x0 ∈ (a, b) be fixed. By Theorem 5.20 and 3.8

c→a+lim



d →b−lim Z d

c

f (x )dx



= lim

c→a+

Z x0 c

f (x )dx + lim

d →b−

Z d x0

f (x )dx



= lim

c→a+

Z x0

c

f (x )dx + lim

d →b−

Z d x0

f (x )dx

= lim

d →b−



c→a+lim Z d

c

f (x )dx

 .2

(22)

### Proof:

Let x0 ∈ (a, b) be fixed. By Theorem 5.20 and 3.8

c→a+lim



d →b−lim Z d

c

f (x )dx



= lim

c→a+

Z x0 c

f (x )dx + lim

d →b−

Z d x0

f (x )dx



= lim

c→a+

Z x0

c

f (x )dx + lim

d →b−

Z d x0

f (x )dx

= lim

d →b−



c→a+lim Z d

c

f (x )dx

 .2

WEN-CHINGLIEN Advanced Calculus (I)

(23)

### Proof:

Let x0 ∈ (a, b) be fixed. By Theorem 5.20 and 3.8

c→a+lim



d →b−lim Z d

c

f (x )dx



= lim

c→a+

Z x0 c

f (x )dx + lim

d →b−

Z d x0

f (x )dx



= lim

c→a+

Z x0

c

f (x )dx + lim

d →b−

Z d x0

f (x )dx

= lim

d →b−



c→a+lim Z d

c

f (x )dx

 .2

(24)

Theorem

If f,g are improperly integrable on (a,b) and α, β ∈R, then αf + βgis improperly integrable on (a,b) and

Z b a

(αf (x ) + βg(x ))dx = α Z b

a

f (x )dx + β Z b

a

g(x )dx .

WEN-CHINGLIEN Advanced Calculus (I)

(25)

Theorem

If f,g are improperly integrable on (a,b) and α, β ∈R, then αf + βgis improperly integrable on (a,b) and

Z b a

(αf (x ) + βg(x ))dx = α Z b

a

f (x )dx + β Z b

a

g(x )dx .

(26)

Theorem (Comparison Theorem For Improper Integrals) Suppose that f,g are locally integrable on (a,b). If

0 ≤ f (x ) ≤ g(x ) for x ∈ (a, b), and g is improperly

integrable on (a,b), then f is improperly integrable on (a,b) and

Z b a

f (x )dx ≤ Z b

a

g(x )dx .

WEN-CHINGLIEN Advanced Calculus (I)

(27)

Theorem (Comparison Theorem For Improper Integrals) Suppose that f,g are locally integrable on (a,b). If

0 ≤ f (x ) ≤ g(x ) for x ∈ (a, b), and g is improperly

integrable on (a,b), then f is improperly integrable on (a,b) and

Z b a

f (x )dx ≤ Z b

a

g(x )dx .

(28)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ). Since f ≥ 0, the function F is increasingon [c,d], hence F (b−) exists.

Thus, by definition, f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

WEN-CHINGLIEN Advanced Calculus (I)

(29)

### Proof:

Fix c ∈ (a, b).Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ).Since f ≥ 0, the function F is increasingon [c,d], hence F (b−) exists.

Thus, by definition, f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

(30)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ). Since f ≥ 0,the function F is increasingon [c,d], hence F (b−) exists.

Thus, by definition, f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

WEN-CHINGLIEN Advanced Calculus (I)

(31)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ).Since f ≥ 0, the function F is increasingon [c,d],hence F (b−) exists.

Thus, by definition, f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

(32)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ). Since f ≥ 0,the function F is increasingon [c,d], hence F (b−) exists.

Thus, by definition, f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

WEN-CHINGLIEN Advanced Calculus (I)

(33)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ). Since f ≥ 0, the function F is increasingon [c,d],hence F (b−) exists.

Thus,by definition, f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

(34)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ). Since f ≥ 0, the function F is increasingon [c,d], hence F (b−) exists.

Thus, by definition,f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

WEN-CHINGLIEN Advanced Calculus (I)

(35)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ). Since f ≥ 0, the function F is increasingon [c,d], hence F (b−) exists.

Thus,by definition, f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

(36)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ). Since f ≥ 0, the function F is increasingon [c,d], hence F (b−) exists.

Thus, by definition,f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

WEN-CHINGLIEN Advanced Calculus (I)

(37)

### Proof:

Fix c ∈ (a, b). Let F (d ) =Rd

c f (x )dx and G(d ) =Rd

c g(x )dx for d ∈ [c, b). By the Comparison Theorem for Integrals, F (d ) ≤ G(d ). Since f ≥ 0, the function F is increasingon [c,d], hence F (b−) exists.

Thus, by definition, f is improper integrable on (c,b) and Z d

c

f (x )dx = F (b−) ≤ G(b−) = Z b

c

g(x )dx . A similar argument works for the case c → a+. 2

(38)

Definition

Let (a,b) be a nonempty, open interval and f : (a, b) →R (i)

f is said to be absolutely integrable on (a,b) if and only if f is locally integrable and |f | is improperly integrable on (a,b).

(ii)

f is said to be conditionally integrable on (a,b) if and only if f is improperly integrable on (a,b).

WEN-CHINGLIEN Advanced Calculus (I)

(39)

Definition

Let (a,b) be a nonempty, open interval and f : (a, b) →R (i)

f is said to be absolutely integrable on (a,b) if and only if f is locally integrable and |f | is improperly integrable on (a,b).

(ii)

f is said to be conditionally integrable on (a,b) if and only if f is improperly integrable on (a,b).

(40)

Definition

Let (a,b) be a nonempty, open interval and f : (a, b) →R (i)

f is said to be absolutely integrable on (a,b) if and only if f is locally integrable and |f | is improperly integrable on (a,b).

(ii)

f is said to be conditionally integrable on (a,b) if and only if f is improperly integrable on (a,b).

WEN-CHINGLIEN Advanced Calculus (I)

(41)

Definition

Let (a,b) be a nonempty, open interval and f : (a, b) →R (i)

f is said to be absolutely integrable on (a,b) if and only if f is locally integrable and |f | is improperly integrable on (a,b).

(ii)

f is said to be conditionally integrable on (a,b) if and only if f is improperly integrable on (a,b).

(42)

Definition

Let (a,b) be a nonempty, open interval and f : (a, b) →R (i)

f is said to be absolutely integrable on (a,b) if and only if f is locally integrable and |f | is improperly integrable on (a,b).

(ii)

f is said to be conditionally integrable on (a,b) if and only if f is improperly integrable on (a,b).

WEN-CHINGLIEN Advanced Calculus (I)

(43)

Definition

Let (a,b) be a nonempty, open interval and f : (a, b) →R (i)

f is said to be absolutely integrable on (a,b) if and only if f is locally integrable and |f | is improperly integrable on (a,b).

(ii)

f is said to be conditionally integrable on (a,b) if and only if f is improperly integrable on (a,b).

(44)

Theorem

If f is absolutely integrable on (a,b), then f is improperly integrable on (a,b) and

Z b a

f (x )dx

≤ Z b

a

|f (x)|dx.

WEN-CHINGLIEN Advanced Calculus (I)

(45)

Theorem

If f is absolutely integrable on (a,b), then f is improperly integrable on (a,b) and

Z b a

f (x )dx

≤ Z b

a

|f (x)|dx.

(46)

### Example:

Prove that the function sin x

x is conditionally integrable on [1, ∞).

WEN-CHINGLIEN Advanced Calculus (I)

(47)

### Example:

Prove that the function sin x

x is conditionally integrable on [1, ∞).

(48)

### Proof:

Integrating by parts, we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞), it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and Z

1

sin x

x dx = cos(1) − Z

1

cos x x2 dx

WEN-CHINGLIEN Advanced Calculus (I)

(49)

### Proof:

Integrating by parts,we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞), it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and

(50)

### Proof:

Integrating by parts, we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞), it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and Z

1

sin x

x dx = cos(1) − Z

1

cos x x2 dx

WEN-CHINGLIEN Advanced Calculus (I)

(51)

### Proof:

Integrating by parts, we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞),it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and

(52)

### Proof:

Integrating by parts, we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞), it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and Z

1

sin x

x dx = cos(1) − Z

1

cos x x2 dx

WEN-CHINGLIEN Advanced Calculus (I)

(53)

### Proof:

Integrating by parts, we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞),it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and

(54)

### Proof:

Integrating by parts, we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞), it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and Z

1

sin x

x dx = cos(1) − Z

1

cos x x2 dx

WEN-CHINGLIEN Advanced Calculus (I)

(55)

### Proof:

Integrating by parts, we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞), it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and

(56)

### Proof:

Integrating by parts, we have Z b

a

sin x

x dx = −cos x x −

d 1

− Z d

1

cos x x2 dx

=cos(1) − cos d

d −

Z d 1

cos x x2 dx Since 1

x2 is absolutely integrable on [1, ∞), it follows from Remark 5.46 that cos x

x2 is absolutely integrable on [1, ∞).

Therefore, sin x

x is improperly integrable on [1, ∞) and Z

1

sin x

x dx = cos(1) − Z

1

cos x x2 dx

WEN-CHINGLIEN Advanced Calculus (I)

(57)

To show that sin x

x is not absolutely integrable on [1, ∞), notice that

Z 1

| sin x|

x dx ≥

n

X

k =2

Z k π (k −1)π

| sin x|

x dx

n

X

k =2

1 k π

Z k π (k −1)π

| sin x|dx

=

n

X

k =2

2

k π = 2

π

n

X

k =2

1 k for each n ∈N

(58)

To show that sin x

x is not absolutely integrable on [1, ∞), notice that

Z 1

| sin x|

x dx ≥

n

X

k =2

Z k π (k −1)π

| sin x|

x dx

n

X

k =2

1 k π

Z k π (k −1)π

| sin x|dx

=

n

X

k =2

2

k π = 2

π

n

X

k =2

1 k for each n ∈N

WEN-CHINGLIEN Advanced Calculus (I)

(59)

To show that sin x

x is not absolutely integrable on [1, ∞), notice that

Z 1

| sin x|

x dx ≥

n

X

k =2

Z k π (k −1)π

| sin x|

x dx

n

X

k =2

1 k π

Z k π (k −1)π

| sin x|dx

=

n

X

k =2

2

k π = 2

π

n

X

k =2

1 k for each n ∈N

(60)

To show that sin x

x is not absolutely integrable on [1, ∞), notice that

Z 1

| sin x|

x dx ≥

n

X

k =2

Z k π (k −1)π

| sin x|

x dx

n

X

k =2

1 k π

Z k π (k −1)π

| sin x|dx

=

n

X

k =2

2

k π = 2

π

n

X

k =2

1 k for each n ∈N

WEN-CHINGLIEN Advanced Calculus (I)

(61)

To show that sin x

x is not absolutely integrable on [1, ∞), notice that

Z 1

| sin x|

x dx ≥

n

X

k =2

Z k π (k −1)π

| sin x|

x dx

n

X

k =2

1 k π

Z k π (k −1)π

| sin x|dx

=

n

X

k =2

2

k π = 2

π

n

X

k =2

1 k for each n ∈N

(62)

To show that sin x

x is not absolutely integrable on [1, ∞), notice that

Z 1

| sin x|

x dx ≥

n

X

k =2

Z k π (k −1)π

| sin x|

x dx

n

X

k =2

1 k π

Z k π (k −1)π

| sin x|dx

=

n

X

k =2

2

k π = 2

π

n

X

k =2

1 k for each n ∈N

WEN-CHINGLIEN Advanced Calculus (I)

(63)

To show that sin x

x is not absolutely integrable on [1, ∞), notice that

Z 1

| sin x|

x dx ≥

n

X

k =2

Z k π (k −1)π

| sin x|

x dx

n

X

k =2

1 k π

Z k π (k −1)π

| sin x|dx

=

n

X

k =2

2

k π = 2

π

n

X

k =2

1 k for each n ∈N

(64)

Since

n

X

k =2

1 k ≥

n

X

k =2

Z k +1 k

1 xdx =

Z n+1 2

1

xdx = log(n+1)−log 2 → ∞ as n → ∞, it follows from the Squeeze Theorem that

n→∞lim Z

1

sin x

x dx = ∞.

Thus, sin x

x is not absolutely integrable on [1, ∞).2

WEN-CHINGLIEN Advanced Calculus (I)

(65)

Since

n

X

k =2

1 k ≥

n

X

k =2

Z k +1 k

1 xdx =

Z n+1 2

1

xdx = log(n+1)−log 2 → ∞ as n → ∞,it follows from the Squeeze Theorem that

n→∞lim Z

1

sin x

x dx = ∞.

Thus, sin x

x is not absolutely integrable on [1, ∞).2

(66)

Since

n

X

k =2

1 k ≥

n

X

k =2

Z k +1 k

1 xdx =

Z n+1 2

1

xdx = log(n+1)−log 2 → ∞ as n → ∞, it follows from the Squeeze Theorem that

n→∞lim Z

1

sin x

x dx = ∞.

Thus, sin x

x is not absolutely integrable on [1, ∞).2

WEN-CHINGLIEN Advanced Calculus (I)

(67)

Since

n

X

k =2

1 k ≥

n

X

k =2

Z k +1 k

1 xdx =

Z n+1 2

1

xdx = log(n+1)−log 2 → ∞ as n → ∞, it follows from the Squeeze Theorem that

n→∞lim Z

1

sin x

x dx = ∞.

Thus, sin x

x is not absolutely integrable on [1, ∞).2

(68)

Since

n

X

k =2

1 k ≥

n

X

k =2

Z k +1 k

1 xdx =

Z n+1 2

1

xdx = log(n+1)−log 2 → ∞ as n → ∞, it follows from the Squeeze Theorem that

n→∞lim Z

1

sin x

x dx = ∞.

Thus, sin x

x is not absolutely integrable on [1, ∞).2

WEN-CHINGLIEN Advanced Calculus (I)

(69)

## Thank you.

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