Trying to figure out this integral

[repoman]

https://www.wolframalpha.com/input/?i=integrate++%28e%5E%283x%29+-e%5Ex%29*%28x*%28e%5Ex%2B1%29*%28e%5E%283x%29%2B1%29%29%5E-1+0+to+infinity

https://www.wolframalpha.com/input/?i=integrate++%28e%5E%283x%29+-e%5Ex%29*%28x*%28e%5Ex%2B1%29*%28e%5E%283x%29%2B1%29%29%5E-1+

It was in an "integration bee" with only four minutes to answer it. Neither guy got the answer. But the answer from 0 to inf is 1/2 ln (3) .

So despite the fact that it is only known by a series and not closed form, what tool box did the question maker expect the contestants to use to be able to get this answer?

If you can't get the definite integral, you can't use L'Hospital's rule, right?

 

[barbos]

contour integration

 

[barbos]

By the way it can be slightly reduced - (exp(x)+1) is a common multiplier there.
And Taylor series is not useful here because it has complex analysis residues at exp(3x)+1=0.
One has to use Loraine series instead.
I suspect that what they were expected to do is to reduce it by (exp(x)+1) and then simply to know the answer from memory, otherwise I see no point in this obfuscation. So it's doable in seconds, especially for trained kids. Doing it without memorizing whole lot of integrals using only first principles is impossible.

reduced form e^x(e^x-1)/(x*(e^(3x)+1))

 

[Tigers!]

Looking at it made my head hurt.

 

[repoman]

An alternative form (from WA) is this

 

[SLD]

0.549306...

But what is that number? It’s not 1/pi or something like that as far as I could tell.

Never mind. 1/2 ln (3).

When I took first year calculus, I recall learning all of these techniques, but sure can’t remember them, now!

 

[fromderinside]

I suggest  CRC Press Standard Mathematical Tables

Won one, CRC Handbook of Chemistry and Physics 39th edition, as a prize for winning best chem student at KHS* in 1957. 'twas a promotional CRC sponsored thing back in the day.

Oh have things changed since then.

*^{_{Kennewick Senior High School, Kennewick WN}}

 

[repoman]

I suggest  CRC Press Standard Mathematical Tables

Won one, CRC Handbook of Chemistry and Physics 39th edition, as a prize for winning best chem student at KHS* in 1957. 'twas a promotional CRC sponsored thing back in the day.

Oh have things changed since then.

*^{_{Kennewick Senior High School, Kennewick WN}}

I think that this is the type of integral that is only closed form defined over that 0 to infinity bound of integration. Any other bounds it will not be.

I guess the integral is still not closed form at 0 and infinity but somehow it should be easily known that for function f(x) the antiderivative F(Infinity)-Fzero) = (1/2)*ln(3). The non closed form terms of the series converge at the bounds.

The is not known from massive computation. .

 

[repoman]

I am trying to do the Feynman trick and needless to say, I don't fully understand how the method works.

First one this is after I slip up the integral to



and the negative of this



Now replace the constants one and three of 1x and 3x in the exponential by variable a.

so f(a) is now the integral



Then you differentiate with respect to a so that

f'(a) =



because the x on top and bottom cancel


anyway more later except now I have to integrate with respect to a to get back to f(a) and then fine tune. And that is assume I did it right and that this trick is even applicable.