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} $
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} $
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.