How Can Double Integration Solve This Tricky Infinite Integral?

[KevinMWHM]

Homework Statement

“Fun Homework Problem” assigned for extra credit...
∫ 0 to ∞ [(e^-x)-(e^-3x)]/x dx



2. Homework Equations (supplied hints)

I can get an abstract answer from wolfram but it’s not how the professor wants us to do it.
He gave us a couple “hints”; try to introduce a second variable and reverse the boundries
for example: ∫∫ f(y) f(x) dy dx? I’m assuming he means integrate with respect to y first.
A second possible hint was to try to get to ∫ 1/x dx by getting -[(e^-x)-(e^-3x)] on top to cancel the other.

The Attempt at a Solution

The biggest problem I’m running into is the x on the bottom.
Can I set new boundries by introducing 1 to y+1? I feel like the only way to do this problem is by getting +n in the denominator in order to prevent division by 0 when evaluating 0 to infinity.
I’ve also tried multipling by x/x in order to get an x^2 and move it to the top but the following integration by parts seems to get me no where.

Is there a trig sub I’m not seeing?

Even my professor admits he’s having trouble with it, but “knows” it’s possible.


Thanks for any help...

 

[arildno]

Scrutinize the integral $\int_{a}^{b}e^{-yx}dy$

 

[jackmell]

The biggest problem I’m running into is the x on the bottom

Hi,

Yeah, that x on the bottom. How about this: I haven't worked this problem yet ok but that's not the point I wish to make. Rather, the essential quality in successful mathematics of trying things and cultivating a tolerance of not getting discouraged when they don't work but rather, motivating yourself into finding other things to try.

Ok, back to that x on the bottom. Well, that reminds me of another problem recently in here:

https://www.physicsforums.com/showthread.php?t=717360

Now, I'm not sure that's even relevant to solving this problem but again, that's not the point I'm trying to make. :)