User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

What to Call this Theorem?
Hello friends, I'd like to add a proof for:


 * $\displaystyle \int \frac {1}{x^2 + a^2} \ \mathrm dx$

but I don't know what to call it. "Integral of a Rational Function"? "Integral Involving Arctangent"? "Integral of 1 Over (x^2 + a^2)"? What should I name the page? --GFauxPas 06:55, 15 December 2011 (CST)

Moved to Integral Involving Arctangent, please change the name if you think of something better. --GFauxPas 14:02, 15 December 2011 (CST)

Moved to Integral Involving Arcsine --GFauxPas 08:26, 16 December 2011 (CST)


 * I appreciate the work you have been doing on these integrals. The only thing bothering me slightly is that you write equations like $\mathrm d x = a \cos\theta \mathrm d\theta$ while I suspect that you do not really understand what this means (the theory of differentials is really quite delicate and technical to deal with formally). Therefore, I suggest you stick with the substitution theorem instead. --Lord_Farin 08:38, 16 December 2011 (CST)
 * Just to muddy the waters, IMO it's okay to write $\mathrm d x = a \cos\theta \mathrm d\theta$ as long as the technique is defined as shorthand for the full derivatives. This would need to be done once on the page definition Integration by Substitution. This would remove the need for all this unsightly and unwieldy work every time it is invoked. Mind, I'm not a teacher (and never will be, officially, for legal reasons) so my view is subservient to LF's.
 * The same would apply to Integration by Parts. There's probably more work to be done on those pages so as to ensure the notation is appropraitely defiend (I havent checked because I'm doing other stuff atm). --prime mover 15:09, 16 December 2011 (CST)

Something like this? This looks... bizarre:

There's probably a more elegant way to do this ... --GFauxPas 11:25, 16 December 2011 (CST)


 * I'd mention Derivative of an Inverse Function to write down $a\cos\theta \frac{\mathrm d \theta}{\mathrm dx} = 1$ separately, and then just plug it in. Maybe by putting that equation in the middle, using $\implies$ to signify your conclusion. This would yield:


 * Hope that makes a bit more sense. --Lord_Farin 11:44, 16 December 2011 (CST)

Okay well in addition to working this issue out, do I need to justify this step, which I plan on using more in the future?:


 * $\left(x/a\right)^2 = x^2/a^2$

For $x \le 0$, I couldn't find a page to link to. --GFauxPas 17:38, 17 December 2011 (CST)