Show that $f: (0,1] \rightarrow \mathbb{R}$ given by $ f(x) = \sin{\frac{1}{x}} $ is Lebesgue-integrable and calculate its Lebesgue Integral

Asked Jun 13 '19 at 17:39

Active Jun 13 '19 at 17:50

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Show that the function $f: (0,1] \to \mathbb{R}$ given by $$ f(x) = \sin{\frac{1}{x}} $$ is Lebesgue-integrable and calculate its Lebesgue Integral.

Here my attempt:

$f$ is continuous in the domain so is Riemann-integrable. but how to calculate $$ \int \sin{\frac{1}{x}}? $$

It seems there is no elementary function for its anti-derivative.

So, how can I obtain $$ \int_{(0,1]} \sin{\frac{1}{x}}? $$

edited Jun 13 '19 at 17:45

gt6989b

asked Jun 13 '19 at 17:39

José Marín

1

Maybe try a substitution such as $u={1 \over t}$. (See Daniel's answer here https://math.stackexchange.com/a/720076/27978.) – copper.hat Jun 13 '19 at 17:43
I already try that and get some strange integral $Ci(x) = \int{\frac{\cos x}{x}}$ – José Marín Jun 13 '19 at 17:48
1

I have not worked through the details, but Daniel's answer is likely correct. – copper.hat Jun 13 '19 at 17:50

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