# Convergence of Improper Integral: Can a Comparison Test Be Used?

• aostraff
In summary, an improper integral is an integral where the upper or lower limit of integration is infinite or has a discontinuity. Its convergence can be determined using various tests such as the comparison test, limit comparison test, or the integral test. A convergent improper integral has a finite limit, while a divergent one does not. Both upper and lower limits of integration can be infinite as long as the limit of the integral exists. The convergence of an improper integral affects its value by either being equal to the limit or being undetermined if it diverges.
aostraff

## Homework Statement

I'm trying to show that this improper integral converges
$$\int_{0}^{1} \sin \left ( x + \frac{1}{x} \right )dx$$

## The Attempt at a Solution

I thought a comparison test would be nice but I can't think of the right one if that is the way to go. I don't think a substitution of u = x + 1/x works too well as well.

Thanks.

-1 <= sin(x + 1/x) <= 1

I see. Thanks.

## 1. What is an improper integral?

An improper integral is an integral where either the upper or lower limit of integration is infinite, or the integrand has a discontinuity at some point within the limits of integration.

## 2. How do you determine if an improper integral converges?

An improper integral converges if the limit of the integral as the upper or lower limit of integration approaches a finite number exists. This can be determined using various tests such as the comparison test, limit comparison test, or the integral test.

## 3. What is the difference between a convergent and a divergent improper integral?

A convergent improper integral is one where the limit of the integral is a finite number, while a divergent improper integral is one where the limit does not exist or is infinite.

## 4. Can an improper integral have both upper and lower limits of integration be infinite?

Yes, an improper integral can have both upper and lower limits of integration be infinite, as long as the limit of the integral as both limits approach a finite number exists.

## 5. How does the convergence of an improper integral affect its value?

If an improper integral converges, its value is equal to the limit of the integral as the limits of integration approach a finite number. If it diverges, its value cannot be determined as the limit does not exist.

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