User talk:Raul654/proof

Before I commit myself to an answer, I am pretty sure the error is in the penultimate step (subtracting the integrals). Am I right here? If I am, I'll give you the full justification on why that step is probably incorrect. Dysprosia 07:47, 17 Jul 2004 (UTC)
 * You are correct. Subtracting the integrals is the flaw in the proof. &rarr;Raul654 08:00, Jul 17, 2004 (UTC)

May I be so bold then, to provide my conclusion:
 * Assume $$\int \tan x = F(x) + C_1$$

Then, in the final step:
 * $$\int \tan x = -1 + \int \tan x $$ (this is fine - if you differentiate both sides you get the desired result)

Now substitute the above - the key thing is we need to use different constants since we don't have assurances that they are the same:
 * $$ F(x) + C_1 = -1 + F(x) + C_2$$

Now we can subtract off F(x) fine:
 * $$ C_1 = -1 + C_2 $$
 * $$ C_1 - C_2= -1 $$

which leads to a consistency - the arbitrary constants can be anything (as they should be), as there are an infinity of solutions to that last equation. Dysprosia 08:11, 17 Jul 2004 (UTC)

You are correct. The constants of integration for the two sides are different. Or, as a friend of mine (who is getting PhD in mathematics) phrased it - the integral of tanget is a set of functions. You can't subtract one set from another like that - algebric manipulation doesn't work on sets. &rarr;Raul654 08:39, Jul 17, 2004 (UTC)


 * Yay :) I'm feeling pretty good about myself right now... Dysprosia 08:57, 17 Jul 2004 (UTC)

I can't believe it!
For about 3 hours during sophomore year of HS, I came up with a very similar proof to this one (albeit simpler) and was convinced I had disproved math until a friend finally pointed out what I had done wrong. Amazing to see that I'm not the only one. Jairuscobb 09:51, 4 August 2007 (UTC)

Another flaw
I believe there's another flaw in this proof. UV substitution is only applicable on continuously differentiable functions. Therefore, you cannot apply this method to this problem. Therefore, $$ \int_ {} \tan (x)\, dx \neq \sec (x) * (- \cos (x)) - \int_ {} - \cos (x) * \sec(x) * \tan(x) \, dx $$ Serpent_Guard 07:24, 30 April 2008 (UTC)


 * Why isn't tan x continuously differentiable? I thought integration by parts could be applied to tan x just as much as it can be applied to tan^2 x anyway. Deamon138 (talk) 21:25, 6 June 2008 (UTC)


 * It's not continuously differentiable because you can't find the derivative of $$\tan(.5*pi)$$.Serpent_Guard 07:24, 12 August 2008 (UTC) —Preceding unsigned comment added by 70.99.191.196 (talk)


 * Ah gotcha, I see what you mean by continuously differentiable now. Deamon138 (talk) 19:07, 12 August 2008 (UTC)