Talk:Integration by Substitution

Is it acceptable to leave the equality as an equality, ignoring that the change of variables might shift what the constant of integration is? Would it be better to put:


 * $\displaystyle \int y\dfrac {\mathrm dy}{\mathrm dx} \, \mathrm dx = \int y \, \mathrm dy + C$

?--GFauxPas (talk) 18:20, 9 January 2013 (UTC)


 * Since the constant is arbitrary, it is meaningless to add it; after all, its value will depend on what you choose for both primitives. Any primitive for the left integrand is a primitive for the right integrand - this justifies equality IMO. A valid point to raise, though. --Lord_Farin (talk) 18:25, 9 January 2013 (UTC)


 * That's what I would have thougt, but I just encountered online a false proof that $1 = 0$, since clearly


 * $\displaystyle \int \frac 1 x \mathrm dx = 1 - \int \frac {-1} x \mathrm dx$


 * using integration by parts, and that makes me a bit nervous. --GFauxPas (talk) 18:26, 9 January 2013 (UTC)


 * No problem. Recall that $\displaystyle \int 0 \, \mathrm dx = C$. --Lord_Farin (talk) 18:28, 9 January 2013 (UTC)


 * What are you trying to emphasize with that? --GFauxPas (talk) 18:39, 9 January 2013 (UTC)


 * Not sure, I'm afraid. --Lord_Farin (talk) 18:49, 9 January 2013 (UTC)