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### Integral Problems

```
Date: 9/14/95 at 7:15:15
From: Filiberto Strazzari
Subject: Generalized Integrals

Let's consider the function

1
f(x) = -----------------
x^h * (log x)^k

For what real values of h and k does

infinite                      1
/                           /
| f(x) dx          and      | f(x) dx
/                           /
1                           0

converge or diverge?

problem. Thank you very much for your effort.

Filiberto Strazzari
<def0721@iperbole.bologna.it>
```

```
Date: 10/11/95 at 17:17:39
From: Doctor Ethan
Subject: Re: Generalized Integrals

Well, here are my thoughts on your problem.

One way to phrase this question is "For what values of h and
k does the bottom grow fast enough so that the integral
converges?"

Let's look at integral one.  Both things on the bottom are
positive for all of these values, so there is no fear that
they will cancel each other out.

When h>1  1/x^h will converge by itself, so adding that extra
stuff for any k will be okay.

So look at h<=1.  You may know that 1/x does not converge,
so one way to think about this is, "When is log[x]^k going
to grow faster than x in the long run? "  We will show that
the answer is that it never will.

The notion of "grow faster" is very closely related to the
derivative, so let us compare the derivative of the two
functions.  The derivative of x is 1 and the derivative of
log[x]^k is k * log[x]^(k-1) *1/x . But as x approaches
infinity this approaches zero.  (You can prove this using
L'Hopital's rule.)

So this tells that for the first one, when h<=1 this will not
converge for any value of k.

Great! Now, on to the second one.

I will just give you a hint and let you see if you can figure
it out.

We know that the polynomial part (i.e. x^h) will have finite
value for h=0 or greater.  So for those you can just examine
possible k values in a similar way to what we did earlier.

Now think about the values for h<0. These start to get
interesting.  Think about them.

Good Luck...

- Doctor Ethan,  The Geometry Forum

```
Associated Topics:
High School Calculus

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