Advanced High-School Mathematics

SECTION 5.1 Quick Survey of Limits 255

The definitions of these improper integrals are in terms of limits.
For example

∫∞

0 f(x)dx = limb→∞

∫b

a f(x)dx

∫∞

−∞f(x)dx = a→−∞lim

∫ 0

a f(x)dx+ limb→∞

∫b

0 f(x)dx.

Likewise, for example,

∫ 1

0

dx

xp

= lim

a→ 0 +

∫ 1

a

dx

xp

,

∫ 3

1

dx

x− 2

= lima→ 2 −

∫a

1

dx

x− 2

+ limb→ 2 +

∫ 3

b

dx

x− 2

.

Relative to the above definition, the following is easy.

Theorem. We have

∫∞

1

dx

xp

=









1

p− 1

ifp > 1

∞ ifp≤ 1.

.

Proof. We have, ifp 6 = 1, that

∫∞

1

dx

xp

= lima→∞

x^1 −p

1 −p

∣∣

∣∣

∣

a

1

= lima→∞

Ñ

a^1 −p

1 −p

−

1

1 −p

é

=











1

p− 1

ifp > 1

∞ ifp < 1.

Ifp= 1, then

∫∞

1

dx

x

= lima→∞lnx

∣∣

∣∣

∣

a

1

= lima→∞lna = ∞.

Example. Compute the improper integral

∫∞

2

dx

xlnpx

, where p >1.