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

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SUMMARY

The integral of (1/x)sin(1/x)dx from 0 to 1 converges absolutely. By applying the comparison test and a change of variables with u = 1/x, the integral transforms to the limits from 1 to infinity. The resulting integral of -sin(u)/u can be evaluated using integration by parts, confirming its convergence. The conclusion is that the integral converges due to the bounded nature of sin(u)/u as u approaches infinity.

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covariance64
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Homework Statement



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


Homework Equations





The Attempt at a Solution



Should we use the comparison test in this situation?
 
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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?


Homework Equations





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
 

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