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I don't really know how I would actually show this. The only thing I can think of is to look at the graph of the function to see that it is convergent. However, how would I do it algebraically?

$ \int_{0}^{1} \frac{\sin(x)dx}{x} $

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    @TenaliRaman: Thanks, will try that.2012-11-20

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The function $\,\dfrac{\sin x}{x}\,$ is continuous and bounded on $\,(0,1]\,$ , and the discontinuity point at zero is removable, so the integral exists.

In fact, this wouldn't usually be considered an improper integral but in fact a proper, definite one.

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    Yes, of course. Thanks!2012-11-20