trouble with an improper integral. can someone please explain to me how my instructor evaluated this integral?

Question

richyw · Answer

So I can work this out but it takes me a LOT of math. how did he do this so easily? like I have no idea how he got the$$\frac{d^2}{d\lambda^2}$$

richyw · Answer

I can follow up to the end of the first line. then BOOM it seems like magic.

tkhunny · Answer

It appears merely to be shorthand for integration by parts applied twice.

richyw · Answer

that would make sense, because that's how I did it. Actually I think I only used one IBP? I can't remember. Either way, I wish I could do math like that in my head!

tkhunny · Answer

Has to be 2 with that x^2 in there.

Ask your professor about it.  Is it applicable ONLY to this type of exponential integral?  Does it have other applications?  It may be a natural reflex because the professor did a research project on this sort of integral.  Return and report!!