“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...