# Partition a divergent integral into finite values

• LikeMath
In summary, the question asks whether an interval with an infinite integral can be partitioned into countable disjoint subintervals with equal integrals, such as 1/2. While there are counterexamples, it is possible for a large class of functions as long as their integrals are well-defined and continuous.
LikeMath
Hi there,

I am reading an article, but I faced the following problem, and I am wondering if it is well known fact.

If the integral of a function on some interval is infinity, can we partition this interval into countable disjoint (in their interiors) subintervals such that the integral on each interval is 1/2 for example?

Likemath

LikeMath said:
Hi there,

I am reading an article, but I faced the following problem, and I am wondering if it is well known fact.

If the integral of a function on some interval is infinity, can we partition this interval into countable disjoint (in their interiors) subintervals such that the integral on each interval is 1/2 for example?

Likemath

It looks like it is doable. You could use any positive finite number. (I assume you mean the integral over each piece = 1/2, so that the total is the sum of an infinite number of 1/2's).

Thank you, but how can I convince my self that it is doable?

Counterexample:
##f:[0,3] \to R##
##f(x)=\frac{1}{1-x}## for ##0\leq x<1##
##f(x)=-1## for ##1\leq x \leq 2##
##f(x)=\frac{1}{x-2}## for ##2 < x \leq 3##

The ugly part is the middle section: You cannot combine it with any interval of the other two sections, as this would diverge.

It is true for a large class of functions, however: if ##\int_a^x f(x') dx'## is well-defined for every x in your interval (a,b), it is continuous and you can find appropriate intervals with the intermediate value theorem. This directly gives a way to count them, too, of course.

## 1. What is a divergent integral?

A divergent integral is an integral that does not have a finite value. This means that the integral either approaches infinity or oscillates between positive and negative infinity.

## 2. Why is it necessary to partition a divergent integral into finite values?

Partitioning a divergent integral into finite values allows us to evaluate the integral and obtain a meaningful result. It also helps us to better understand the behavior of the integral and make comparisons between different integrals.

## 3. How do you partition a divergent integral into finite values?

One method is to split the integral into smaller parts, where each part has a finite value. This can be done by using properties of integrals, such as linearity and the fundamental theorem of calculus. Another method is to use a change of variables, which transforms the integral into a form that can be evaluated.

## 4. Can all divergent integrals be partitioned into finite values?

No, not all divergent integrals can be partitioned into finite values. Some integrals are inherently divergent and cannot be evaluated using traditional methods.

## 5. How is partitioning a divergent integral related to convergence?

Partitioning a divergent integral into finite values is a way to make the integral converge. By breaking the integral into smaller parts, each with a finite value, we can add them together to obtain a finite result. This process is known as regularization of a divergent integral.

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