## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 36 analysis/integral.p

If f is integrable on (0,inf) and differentiable at 0, and a > 0, and:

inf f(x)
Int ---------------- dx is defined
0 x

show:

inf ( f(x) - f(ax) )
Int ---------------- dx = f(0) ln(a)
0 x

analysis/integral.s

First, note that if f(0) is 0, then by substituting u=ax in

the integral of f(x)/x, our integral is the difference of two

equal integrals and so is 0 (the integrals are finite because f is

0 at 0 and differentiable there. Note I make no requirement of

continuity).

Second, note that if f is the characteristic function of the

interval [0, 1]--- i.e.

1, 0<=x<=1
f (x) =
0 otherwise

then a little arithmetic reduces our integral to that of

1/x from 1/a to 1 (assuming a>1; if a <= 1 the reasoning is similar),

which is ln(a) = f(0)ln(a) as required. Call this function g.

Finally, note that the operator which takes the function f to the

value of our integral is linear, and that every function meeting the

hypotheses (incidentally, I should have said `differentiable from the right',

or else replaced the characteristic function of [0,1] above by that of

(-infinity, 1]; but it really doesn't matter) is a linear combination of

one which is 0 at 0 and g, to wit

f(x) = f(0)g(x) + (f(x) - g(x)f(0)).

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