192
CHAPTER 23
23.60
Prove that
for any distinct a, b in [1, +<»).
Case 1.
By the Mean-Value Theorem,
for some c in (a, b). Since
Hence,
and, therefore,
In
Case 2.
By Case 1,
But
and
23.61 Evaluate
23.62
Find
In
In
In
In
23.63
Evaluate
In
In
In 3 - In 2,
23.64 Prove the basic property of logarithms: \nuv = lnu+ \nv.
In
make the change of variable w = ut (u fixed). Then dw = udt and the limits of
integration t = I and t = v go over into w = u and w = uv, respectively. Hence,
So,
In
In
In
In
23.65
If a is a positive constant, find the length of the curve
In x between x = l and x = 2.
So, the length
23.66
If a and b are positive, find the arc length of
In
from
to
Hence,
So, the arc length
In
In
In
In
In
x=a
x=3a.