# Prove Convergence of Integral (1/x)sin(1/x)dx from 0 to 1

• covariance64
In summary, the conversation is discussing how to prove the absolute convergence of the integral of (1/x)sin(1/x)dx from 0 to 1. The suggestion is to use the comparison test or try a change of variables, u=1/x, to simplify the integral. After using integration by parts, it is determined that the integral of sinu/u is always convergent.
covariance64

## Homework Statement

Show that the integral of (1/x)sin(1/x)dx from 0 to 1 is converges absolutely?

## The Attempt at a Solution

Should we use the comparison test in this situation?

Try a change of variables first.

covariance64 said:

## Homework Statement

Show that the integral of (1/x)sin(1/x)dx from 0 to 1 is converges absolutely?

## The Attempt at a Solution

Should we use the comparison test in this situation?

change of variable u=1/x you will get from 1 to infinite integral of -sinu over u after that you do integration by parts and you will get your answer sinu / u is always convergent

## What is the definition of convergence?

The concept of convergence refers to the behavior of a sequence or series as its terms approach a certain limit or become infinitely large. In the context of integrals, convergence refers to the idea that the value of the integral approaches a finite value as the limits of integration (in this case, 0 and 1) become infinitely close together.

## What is the significance of proving convergence?

Proving convergence is important because it ensures that the value of the integral is well-defined and can be accurately calculated. Without convergence, the integral may not have a finite value and therefore cannot be calculated using traditional methods.

## How can the convergence of an integral be proven?

There are several methods for proving the convergence of an integral, including the comparison test, the ratio test, and the limit comparison test. In this specific case, we can use the limit comparison test by comparing the given integral to a known convergent integral, such as 1/x.

## What is the role of the function sin(1/x) in this integral?

The function sin(1/x) plays a crucial role in determining the convergence of the integral. Unlike 1/x, which is a continuous function, sin(1/x) is a discontinuous function with infinitely many discontinuities in the interval [0, 1]. This fact makes the integral more challenging to analyze and requires a different approach for proving convergence.

## What is the result of proving the convergence of this integral?

The result of proving the convergence of this integral is that we can now accurately calculate its value using traditional methods. Additionally, it allows us to make important conclusions about the behavior of the function sin(1/x) and its relationship to the integral as x approaches 0.

Replies
5
Views
2K
Replies
13
Views
2K
Replies
3
Views
755
Replies
5
Views
1K
Replies
2
Views
1K
Replies
7
Views
1K
Replies
11
Views
2K
Replies
16
Views
3K
Replies
11
Views
935
Replies
23
Views
1K