Advanced High-School Mathematics

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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.
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