Note that
and use the weighted mean value theorem (Theorem 3.16 in Apostol) to prove that for positive :
Using compute the integral to six decimal places.
Recall the weighted mean value theorem:
For functions and
continuous on
, if
never changes sign in
then there exists
such that
Proof. Let
Then,
for some . Since
is strictly decreasing on
, we know
; thus,
Furthermore,
Thus,
Now, taking we compute,
there is an error it must be f(c) not 1/c