Some math problems to play with!

Please see below:

 

[Update: Please do not work on this problem. I overlooked something, and now it's fixed. The premise of the question now is false, so do not attempt it!]

The following is the part of a bigger problem I am working on. At one point I've to integrate [5/(3*e^x - 2), x]. Upon integrating, I get the following solution:

integrate[5/(3*E^x - 2), x] ==
(-5*(x - Log[-2 + 3*E^x]))/2

However, is it not, by applying the basic mathematics rule, possible to manipulate the above function and get this rather simpler solution?

integrate[(5 * E^(-x))/((3*E^x - 2) *E^(-x)), x]==

(-5*(x - Log[-2 + 3*E^x]))/2

what is the basic mathematics rule you are using to manipulate?

 How are you adding 2 in both denominator and numerator in the second line? I guess I am not getting your question because I know you can not make such a silly mistake...

Thanks, Marconi. I finally figured out my problem. When I run into messy problems like those, I simply do not see the simpler mistakes.

Thanks again. :)

I was wrong, F22! Thanks for looking into the problem. :)