Is the double integral exists for $ f(x,y) = \frac{e^{-y} \sin x}{x} $

Asked Jan 08 '20 at 12:36

Active Jan 08 '20 at 13:58

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Define the function: $\forall (x,y) \in \mathbb{R}^{*} \times \mathbb{R} $,

$$ f(x,y) = \frac{e^{-y} \sin x}{x} $$ where $\mathbb{R}^{*}$ is the set of real numbers without zero.

$1.$ Evaluate "if exist" the double integral for the function defined over the set $S = (0, n) \times (0, \infty) $.

$2.$ Then compare it with the double integral of the function, defined on $\mathbb{R} \times \mathbb{R}$, $h(x,y) = e^{-xy} \sin x$

I am suffering from this question and really I don't know how to deal with it. Any help please?

The integral of a similar function with one variable was discussed in this question but the solution doesn't help in my case.

edited Jan 08 '20 at 13:58

asked Jan 08 '20 at 12:36

Noah16

You take the double integral, and note that you can split it up into a product of two integrals, both of which work. – Lukas Rollier Jan 08 '20 at 12:40
Fubini's theorem – Lukas Rollier Jan 08 '20 at 12:58
In the second, you can just integrate $y$ out of there, treating $x$ as a constant, and then working with the rest – Lukas Rollier Jan 08 '20 at 13:14
What do you mean by $\mathbb R^*$? – Math1000 Jan 08 '20 at 13:38
@Math1000 the set of real numbers without zero – Noah16 Jan 08 '20 at 13:43

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