Advanced Calculus (I)
WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
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)
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
.
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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
.
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)
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
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)
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
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)
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
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)
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 .
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)
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 .
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)
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
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)
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
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)
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
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)
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
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)
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
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)
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).
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)
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).
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)
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).
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)
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.
Example:
Prove that the function sin x
x is conditionally integrable on [1, ∞).
WEN-CHINGLIEN Advanced Calculus (I)
Example:
Prove that the function sin x
x is conditionally integrable on [1, ∞).
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)
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
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)
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
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)
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
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)
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
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)
To show that sin x
x is not absolutely integrable on [1, ∞), notice that
Z nπ 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
To show that sin x
x is not absolutely integrable on [1, ∞), notice that
Z nπ 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)
To show that sin x
x is not absolutely integrable on [1, ∞), notice that
Z nπ 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
To show that sin x
x is not absolutely integrable on [1, ∞), notice that
Z nπ 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)
To show that sin x
x is not absolutely integrable on [1, ∞), notice that
Z nπ 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
To show that sin x
x is not absolutely integrable on [1, ∞), notice that
Z nπ 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)
To show that sin x
x is not absolutely integrable on [1, ∞), notice that
Z nπ 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
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 nπ
1
sin x
x dx = ∞.
Thus, sin x
x is not absolutely integrable on [1, ∞).2
WEN-CHINGLIEN Advanced Calculus (I)
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 nπ
1
sin x
x dx = ∞.
Thus, sin x
x is not absolutely integrable on [1, ∞).2
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 nπ
1
sin x
x dx = ∞.
Thus, sin x
x is not absolutely integrable on [1, ∞).2
WEN-CHINGLIEN Advanced Calculus (I)
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 nπ
1
sin x
x dx = ∞.
Thus, sin x
x is not absolutely integrable on [1, ∞).2
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 nπ
1
sin x
x dx = ∞.
Thus, sin x
x is not absolutely integrable on [1, ∞).2
WEN-CHINGLIEN Advanced Calculus (I)
