Giáo trình tích phân suy rộng Lê Xuân Đại ĐH Bách Khoa


+∞
a
f (x )dx
f (x) ∀x  a
[a, b]. [a, +∞)
Φ(b) =

b
a
f (x)dx.
I = lim
b→+∞
Φ(b) = lim
b→+∞

b
a
f (x)dx
f (x) [a, +∞)
+∞

a
f (x)dx.

+∞
a
f (x )dx
I = lim
b→+∞


b
a
f (x)dx
I ∞
f (x)  0, ∀x ∈ [a, +∞),
x = a, Ox f (x)
f (x)  0, ∀x ∈ [a, +∞)
0
lim
x→+∞
f (x) = A = 0,
f (x) [a, b] ⊂ [a, +∞)

+∞
a
f (x)dx

b
−∞
f (x )dx
f (x) ∀x  b
[a, b]. (−∞, b]
Ψ(a) =

b
a
f (x)dx.
I = lim
a→−∞

Ψ(a) = lim
a→−∞

b
a
f (x)dx
f (x) (−∞, b]
b

−∞
f (x)dx.

b
−∞
f (x )dx
I = lim
a→−∞

b
a
f (x)dx
I ∞
f (x)  0, ∀x ∈ (−∞, b ],
x = b, Ox f (x)

+∞
−∞
f (x) R
[a, b] ∀c ∈ R
f (x) (−∞, +∞)


+∞
−∞
f (x)dx =

c
−∞
f (x)dx +

+∞
c
f (x)dx
f (x) F (x)
[a, +∞) [a, b].

+∞
a
f (x)dx
lim
b→+∞
F (b) = F (+∞).

+∞
a
f (x)dx = F (+∞) − F (a) = F (x)



+∞
a

.

b
−∞
f (x)dx = F (b) − F (−∞) = F (x)



b
−∞
.
b

−∞
f (x)dx
lim
a→−∞
F (a) = F (−∞)

+∞
−∞
f (x)dx =

F (c) − lim
a→−∞
F (a)

+
+


lim
b→+∞
F (b) − F (c)


+∞
−∞
f (x)dx
lim
a→−∞
F (a)
lim
b→+∞
F (b)

+∞
−∞
f (x)dx = F (+∞) − F (−∞) = F (x)



+∞
−∞
.
I =

+∞
0
cos xdx.
I = sin x




+∞
0
= lim
b→+∞
sin b − sin 0 = lim
b→+∞
sin b.
I
I =

+∞
0
cos xdx.
I = sin x



+∞
0
= lim
b→+∞
sin b − sin 0 = lim
b→+∞
sin b.
I
I =


−1
−∞
dx
x
2
I = −
1
x




−1
−∞
= 1 + lim
a→−∞
1
a
= 1.
I
I =

−1
−∞
dx
x
2
I = −
1
x





−1
−∞
= 1 + lim
a→−∞
1
a
= 1.
I
I =

+∞
−∞
dx
1 + x
2
I = arctan x



+∞
−∞
= lim
b→+∞
arctan b− lim
a→−∞
arctan a =

=
π
2
−

−
π
2

= π.
I
I =

+∞
−∞
dx
1 + x
2
I = arctan x



+∞
−∞
= lim
b→+∞
arctan b− lim
a→−∞
arctan a =
=

π
2
−

−
π
2

= π.
I
I =

+∞
0
xe
−x
2
dx
I = −
1
2
e
−x
2




+∞
0

= lim
b→+∞
−
1
2
e
−b
2
+
1
2
=
1
2
I
I =

+∞
0
xe
−x
2
dx
I = −
1
2
e
−x
2





+∞
0
= lim
b→+∞
−
1
2
e
−b
2
+
1
2
=
1
2
I
I =

+∞
a
dx
x
α
, a > 0, α ∈ R
α = 1
I = −

1
α − 1

lim
x→+∞
1
x
α−1
−
1
a
α−1

α > 1 lim
x→+∞
1
x
α−1
= 0 I =
a
1−α
α − 1
.
I
I =

+∞
a
dx
x

α
, a > 0, α ∈ R
α = 1
I = −
1
α − 1

lim
x→+∞
1
x
α−1
−
1
a
α−1

α > 1 lim
x→+∞
1
x
α−1
= 0 I =
a
1−α
α − 1
.
I
α < 1 lim
x→+∞

1
x
α−1
= +∞ I
α = 1 I = lim
x→+∞
ln |x| − ln a = +∞
I
α > 1 I =
+∞

a
dx
x
α
α  1 I =
+∞

a
dx
x
α
α < 1 lim
x→+∞
1
x
α−1
= +∞ I
α = 1 I = lim
x→+∞

ln |x| − ln a = +∞
I
α > 1 I =
+∞

a
dx
x
α
α  1 I =
+∞

a
dx
x
α
α < 1 lim
x→+∞
1
x
α−1
= +∞ I
α = 1 I = lim
x→+∞
ln |x| − ln a = +∞
I
α > 1 I =
+∞

a

dx
x
α
α  1 I =
+∞

a
dx
x
α