# Double improper integral , how to see if it diverge

$$\iint_D \frac{(x+y) e^{y-x}}{x^2 y^2}dx \, dy$$

$$D= \{(x,y) ; 0\leq y+1\leq x , xy\geq 1 \}$$

Iv been stuck on this for past two hours , I need some hint .

My bounds are : $\frac{1+\sqrt{5}}{2}\leq X<\infty$

$\frac 1 x \leq Y\leq x-1$ are the bounds correct ?

I need some hints, Thanks in advance

• So far I can only confirm that the bound of your domain are indeed correct Commented Nov 7, 2016 at 22:20
• Thanks for taking your time , we just started on this topic and I don't know how to proceed , integrating with respect to x and y both fails. Commented Nov 7, 2016 at 22:25
• My next idea is to try comparison test but what should I campare it with ? Commented Nov 7, 2016 at 22:26
• I've deleted my incomplete answer, but I'm leaning toward applying a comparison test after massaging the integral a bit. $\qquad$ Commented Nov 8, 2016 at 1:26
• That was my idea too but I could not find any function to compare with that diverges . Commented Nov 8, 2016 at 16:15

I think this integral converges. Let $a= (1+\sqrt 5)/2.$ Note that $(x+y)/(x^2y^2) = 1/(xy^2) + 1/(x^2y).$ Let's look at the integral that involves the first of these terms. That equals

$$\tag 1 \int_a^\infty \frac{e^{-x}}{x} \int_{1/x}^{x-1}\frac{e^y}{y^2}\,dy \, dx.$$

The inner integral is bounded above by

$$\tag 2 \int_{1/x}^{1}\frac{e^y}{y^2}\,dy + \int_{1}^{x}\frac{e^y}{y^2}\,dy.$$

The first integral in $(2)$ is bounded above $e(x-1).$ If we insert that into the outer integral in $(1),$ we get a convergent integral. For the second integral in $(2)$ note that by L'Hopital,

$$\frac{\int_1^x (e^y/y^2)\, dy}{e^x/x^2} \to 1$$

as $x\to \infty.$ So this integral is $\le 2e^x/x^2$ for large $x.$ Inserting that term into the outer integral in $(1)$ also gives a convergent integral.

That takes care of the part of the original integral involving $1/(xy^2).$ The part involving $1/(x^2y)$ can be handled the same way.

• I will take a look right now at your solution , but am very sure that the i wrote the question right , ie diverges , i looked it up again , that been being said my teacher could made a mistake. Thanks for your answer . Commented Nov 8, 2016 at 17:31
• If i can show that $e^{y-x}$ diverges and what we multiply it by converges , is that considered to be a legal step ? is the product of divergent and convergent = divergent ? Commented Nov 8, 2016 at 17:55
• Of course not. In fact divergent times divergent can be convergent ($1/x$ on $[1,\infty)$ for example)
– zhw.
Commented Nov 8, 2016 at 18:00
• do you know where i can find the rules such as , sum and product of convergent and divergent series ,and what the aritmatic on them? thanks Commented Nov 8, 2016 at 18:14
• Not sure why you're looking to me for a proof of divergence, when I think the integral converges.
– zhw.
Commented Nov 8, 2016 at 19:44

HINT

Try $u = xy$

$v=y-x$

Everything will simplfy with the Jacobian.