LECTURE 22: IMPROPER INTEGRALS
MINGFENG ZHAO
March 04, 2015
Theorem 1. Let f be a forth order differentiable function on [a, b], EM , ET and ES be the absolute errors corresponding
Z b
to the Midpoint, Trapezoid and Simpson’s Rule approximation of
f (x) dx using n subintervals. Assume that |f 00 (x)| ≤
a
k and |f (4) (x)| ≤ K for all a ≤ x ≤ b, then
EM ≤
k(b − a)3
,
24n2
ET ≤
k(b − a)3
,
12n2
ES ≤
and
K(b − a)5
if n is even.
180n4
1
Z
Example 1. Estimate the absolute errors when approximating
sin(x) dx by using Midpoint, Trapezoid and Simp0
son’s Rules with n = 10 subintervals.
Let f (x) = sin(x), then
f 0 (x) = cos(x),
f 00 (x) = − sin(x),
f (3) (x) = − cos(x),
and f (4) (x) = sin(x).
Then we know that
|f 00 (x)| ≤ 1,
and |f (4) (x)| = | sin(x)| ≤ 1,
for all x ∈ [0, 1].
So we have
EM (10) ≤
1
1 · (1 − 0)3
=
,
2
24 · 10
2400
ET (10) ≤
1 · (1 − 0)3
1
=
,
2
12 · 10
1200
and ES(10) ≤
Improper integrals for infinite intervals
Definition 1. Let a and b be two real numbers, then
I. If f is continuous on [a, ∞), then
∞
Z
b
Z
f (x) dx := lim
a
II. If f is continuous on (−∞, b], then
Z
b→∞
b
f (x) dx.
a
Z
f (x) dx := lim
−∞
a→−∞
1
b
f (x) dx.
a
1 · (1 − 0)5
1
=
.
4
180 · 10
1800000
2
MINGFENG ZHAO
III. If f is continuous on (−∞, ∞) and for any real number c, then
Z
∞
Z
a→−∞
Z
b
f (x) dx + lim
f (x) dx := lim
−∞
c
b→∞
a
f (x) dx.
c
For the integrals in the above three cases, if the integral is finite, we say the integral converges; otherwise, the integral
diverges.
∞
Z
e−3x dx.
Example 2. Evaluate
0
In fact, we have
Z
∞
e−3x dx
Z
=
0
b
lim
b→∞
e−3x dx
0
b
1 −3x = lim − e
b→∞
3
0
1 1 −3b
= lim
− e
b→∞
3 3
=
Z
∞
Example 3. Evaluate
−∞
1
.
3
dx
.
1 + x2
In fact, we have
Z
∞
−∞
dx
1 + x2
lim
Z b
dx
dx
+ lim
2
2
b→∞
a 1+x
1 1+x
1
b
tan−1 a + lim tan−1 1
lim
−1
tan (1) − tan−1 (a) + lim tan−1 (b) − tan−1 (1)
Z
=
lim
a→−∞
=
a→−∞
=
a→−∞
1
b→∞
b→∞
=
π π
tan−1 (1) + + − tan−1 (1)
2
2
=
π.
1
Example 4. Let p be a real number and f (x) = p . for what values of p does
x
For any b > 1, we have
Z
b
Z
f (x) dx =
1
1
b
1
dx =
xp
Z
1
b
x−p
Z
∞
f (x) dx converges?
1


 ln(b),
dx =
1


· b1−p − 1 ,
1−p
if p = 1,
if p 6= 1.
LECTURE 22: IMPROPER INTEGRALS
3
Notice that lim ln(b) = ∞ and
b→∞



0, if α < 0,



α
lim b =
1, if α = 0,

b→∞



 ∞, if α > 0.
So we get



∞,
if p = 1,


Z b
 1
lim
f (x) dx =
, if p > 1,

b→∞
p−1
1



 ∞,
if p < 1.
∞
Z
Therefore, we know that
f (x) dx converges if and only if p > 1.
1
Improper integrals for unbounded integrands
Definition 2. Let a and b be two real numbers, then
I. If f is continuous on (a, b] with lim f (x) = ∞ or −∞, then
x→a+
b
Z
Z
f (x) dx := lim+
f (x) dx.
c→a
a
b
c
II. If f is continuous on [a, b) with lim f (x) = ∞ or −∞, then
x→b−
Z
a
b
Z
f (x) dx := lim−
c→b
c
f (x) dx.
a
III. Let a < p < b, if f is continuous on (a, b] with lim f (x) = ∞ or −∞, then
x→p
Z
a
b
Z
f (x) dx := lim−
c→p
a
c
Z
f (x) dx + lim+
d→p
b
f (x) dx.
d
For the integrals in the above three cases, if the integral is finite, we say the integral converges; otherwise,
the integral diverges.
Z
3
Example 5. Evaluate
√
−3
In fact, we have
Z 3
1
√
dx
9
− x2
−3
1
dx.
9 − x2
Z b
1
1
√
dx + lim
dx
−
2
b→3
9
−
x
9
− x2
a
1
x 1
x b
sin−1
+ lim− sin−1
3 a b→3
3 1
Z
=
=
lim
a→−3+
lim
a→−3+
1
√
4
MINGFENG ZHAO
1
b
1
−1
−1 a
−1
−1
=
lim
+ lim sin
sin
− sin
− sin
+
−
3
3
3
3
a→−3
b→3
1
1
= sin−1
− sin−1 (−1) + sin−1 (1) − sin−1
3
3
π π
+
=
2
2
= π.
10
Z
Example 6. Evaluate
dx
1
1
(x − 2) 3
.
In fact, we have
Z
10
1
a
Z
dx
(x − 2)
=
1
3
lim−
Z
dx
1
3
+ lim+
10
dx
1
b→2
(x − 2)
(x − 2) 3
b
a
10
2
2
3
3
= lim− (x − 2) 3 + lim+ (x − 2) 3 2
2
b→2
a→2
1
b
2
2
3
3
3 2
3
= lim−
(a − 2) 3 −
· 8 3 − (b − 2) 3
+ lim+
2
2
2
2
a→2
b→2
a→2
1
3 3
= − + ·4
2 2
9
=
.
2
Example 7. Let p be a real number and f (x) =
1
. for what values of p does
xp
Z
1
f (x) dx converges?
0
For any 0 < a < 1, we have
Z
1
Z
f (x) dx =
a
a
1
1
dx =
xp
Z
a
1
x−p


 − ln(a),
if p = 1,
dx =
1


· 1 − a1−p , if p 6= 1.
1−p
So we get



∞,
if p = 1,


Z 1
 1
lim
f (x) dx =
, if p < 1,

1−p
a→0+
a



 −∞,
if p > 1.
Z
Therefore, we know that
1
f (x) dx converges if and only if p < 1.
0
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca