Is this integral convergent or divergent?

In summary, the given integral is convergent and can be evaluated using a "completion of squares" technique. The comparison theorem may not be applicable, and evaluating the original integral itself would be a better approach for proof.
  • #1
fk378
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0

Homework Statement


Determine whether the following integral is convergent or divergent. If convergent, what does it converge to?

dx/(4x^2 + 4x + 5) [-infinity, infinity]

Homework Equations


comparison theorem?

The Attempt at a Solution


I think it is convergent, so I set the original integral less than or equal to dx/4x^2.
Solving the integral and setting up a limit, I got the limit as t-->infinity of (4x^-1)/-1 evaluated from [-infinity, t]. Now here is where I get lost. Evaluating it at t gives 0, but when I plug in the lower limit of 0, isn't it possible to say that the 0 either belongs in the numerator if there is a -1 exponent there, as well as saying that it doesn't exist if I move (4x^-1) into the denominator?
 
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  • #2
fk378 said:
I think it is convergent, so I set the original integral less than or equal to dx/4x^2.
Solving the integral and setting up a limit, I got the limit as t-->infinity of (4x^-1)/-1 evaluated from [-infinity, t]. Now here is where I get lost. Evaluating it at t gives 0, but when I plug in the lower limit of 0, isn't it possible to say that the 0 either belongs in the numerator if there is a -1 exponent there, as well as saying that it doesn't exist if I move (4x^-1) into the denominator?

I think you want to be careful about comparing your integral to the integral for (4x)^(-2), because that function is not continuous at x = 0 , so you can't use the Fundamental Theorem on it and its integral doesn't converge as x approaches 0.

For the purpose of proof, you may be better off evaluating the original integral itself, since that integrand is continuous. You are correct that it is convergent and can be evaluated using a "completion of squares" technique (since the quadratic is irreducible).
 
  • #3
So if I want to plug in 0 into (4x^-1) then I have to move the 4x to the bottom instead of keeping it at the top with the -1 exponent?
 
  • #4
fk378 said:
So if I want to plug in 0 into (4x^-1) then I have to move the 4x to the bottom instead of keeping it at the top with the -1 exponent?

It's all the same: the point is that this is 1/(4x), so putting zero in for x will make this undefined. The type II improper integral about x = 0 for dx/(4x^2) diverges, so it's no help in making a comparison with your integral.

Fortunately, the type I improper integral you are given is not particularly difficult to evaluate.
 

1. What is "convergence" and "divergence" in science?

Convergence and divergence are terms used to describe the behavior of a series, which is a sequence of numbers that are added together. Convergence refers to a series that approaches a finite value as more terms are added, while divergence refers to a series that either approaches infinity or does not approach any specific value.

2. How can I determine if a series is convergent or divergent?

There are several tests that can be used to determine if a series is convergent or divergent, such as the comparison test, the ratio test, or the integral test. These tests involve evaluating the behavior of the terms in the series and comparing them to known convergent or divergent series.

3. What are some real-life applications of convergence and divergence?

Convergence and divergence have many applications in various fields of science, including physics, economics, and biology. For example, in physics, these concepts are used to study the behavior of infinite series in quantum mechanics and general relativity. In economics, they are used to model the growth of populations or markets. In biology, they are used to analyze the evolution of species.

4. Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent. If a series approaches a finite value as more terms are added, it is convergent. If a series either approaches infinity or does not approach any specific value, it is divergent.

5. What is the importance of understanding convergence and divergence in scientific research?

Understanding convergence and divergence is crucial in scientific research as it allows us to make accurate predictions and draw meaningful conclusions from data. It also helps us determine the validity and reliability of mathematical models and theories. Additionally, being able to identify and analyze convergent and divergent series can lead to new discoveries and advancements in various fields of science.

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