User talk:Raul654/proof

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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. →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) + C1 = − 1 + F(x) + C2

Now we can subtract off F(x) fine:

C1 = − 1 + C2
C1 − C2 = − 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. →Raul654 08:39, Jul 17, 2004 (UTC)

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