Integral of derivative inconsistent

[dc20]

Hi,
I ran into a situation I haven't experienced before where the integral of the derivative doesn't get me the original equation.
Is there a fundamental principle I am missing with this specific example ?
Derivative of this...
(x^2) / ( (x^2)-4 )
...gives...
(-8*x) / ( (x^2-4)^2 )
But integral of (that)...
(-8*x) / ( (x^2-4)^2 )
...gives...
(4) / ( (x^2-4) )
...instead of (x^2) / ( (x^2)-4 ) (original equation).

 

[Hurkyl]

That's because you have your integral wrong! Remember that little extra bit that you're supposed to include, but never bother? ...

 

[dc20]

If you're referring to the constant that accompanies an integral answer, I don't see how that gets me an "x" in the numerator. That would just give me

(4) / ( (x^2-4) ) + C

(I'm doing the derivative first)

Another hint please ?

 

[Hurkyl]

(1) Have you tried doing anything at all with the expression (4) / ( (x^2-4) ) + C ?


(2) Try stating mathematically what you would like to happen.


Either of these by itself should lead you to something that would make you happy. Have you tried either yet? These are the sorts of things that you should have already thought of -- get into that habit if you aren't there yet!

 

[dc20]

ok, I see: C = ( (x^2-4) ) / ( (x^2-4) ) = 1

giving in the numerator 4 + (x^2-4) = x^2 , resulting in the original equation (x^2) / ( (x^2)-4 ).


thanks, but I think your attitude or at least the way you come across in text could use an adjustment. The condescending comments aren't needed. Your comment was edited here, but came thru in the email and I can tell you I am far from lazy.

 

[Hurkyl]

I tend to have a rather silly attitude -- I edited it once I realized that it would be interpreted as condascending (which I had not intended) instead of general silliness. Sorry if it bothered you.

 

[dc20]

no problem. thanks again for your help.