This is easy, but I couldn't find some example of a function that is not integrable but its Riemann improper integral exists and is finite
Needing an example of one riemann integrable function
0
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functional-analysis
measure-theory
vector-analysis
lebesgue-integral
2 Answers
6
The standard example is $f(x)={\sin(x)\over x}$. The integral of the positive part diverges by comparison with the harmonic series, while the improper Riemann integral exists by use of the alternating series theorem.
3
Let $f(x) := \frac{\sin x}{x}$, then
$\int_{\mathbb{R}} f(x) \, d\lambda(x)$
(where $\lambda$ denotes the Lebesgue-Measure) does not exist, whereas the improper Riemann integral
$\int_{\mathbb{R}} f(x) \, dx$
exists (and is finite).