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In the theorem of the book I'm using it states that bijectivity of the substitution function is sufficient. But isn't just locally bijective enough?

Ex:

$\int \frac{x+1}{x^{1/2}}dx=\int \frac{t^2+1}{|t|}dt = \left\{ \begin{array}{lr} \int \frac{t^2+1}{t} & : t>0\\ \int \frac{t^2+1}{-t} & : t<0 \end{array} \right. =2\frac{x^{1/3}}{3}+2x^{1/2}+C$, for $x=t^2 \neq 0$. So, according to my book, $t$ should belong to either the nonnegative reals or nonpositive reals. But from the calculations above, if t belonged to the reals except 0, no harm would come to this world =)... I think.

Am I correct?

Any help would be appreciated.

An old man in the sea.
  • 5,241
  • "According to my book, $t$ should belong to either the nonnegative reals or nonpositive reals". What does your book understand by $\int f(x)\mathrm dx=g(x)$? If your computations are correct, then differentiating $x\mapsto 2\frac{x^{1/3}}{3}+2x^{1/2}+C$ yields $x\mapsto \dfrac{x+1}{\sqrt x}$. If your computations are correct, then this is simply true. So what do you understand by $\int f(x)\mathrm dx=g(x)$? Edit: Bijectivity isn't necessary, it is sufficient. – Git Gud Dec 04 '14 at 11:17
  • @GitGud The $\int f(x)dx$ is the set of all functions whose derivative equal $f(x)$. So, I'm editing my question. – An old man in the sea. Dec 04 '14 at 11:24
  • @GitGud Now, the question is if locally bijective is enough to have a sufficient condition. – An old man in the sea. Dec 04 '14 at 11:26
  • I'm taking you understand that $g(x)=\int f(x)\mathrm dx$ actually means that $g$ is in the set $\int f(x)\mathrm dx$. I am guessing that in the text you're following the theory is restricted to intervals, so $\int f(x)\mathrm dx$ would be the set of all functions whose derivative equals $f$ and whose domain is an interval. Can you confirm this? If this is the case, then that's why they consider only $]-\infty ,0[[$ and $]0,+\infty[$. – Git Gud Dec 04 '14 at 11:30
  • You don't need bijectivty of any kind when talking just about intervals. See Relation to the fundamental theorem of calculus. – Git Gud Dec 04 '14 at 11:32
  • @GitGud Thanks! ;) – An old man in the sea. Dec 04 '14 at 11:40
  • @GitGud Sorry for commenting again. But my book, also has that theorem for integration. However, how does one find the interval of integration if the transformation used is not injective in that interval(without the upper and lower bounds)? – An old man in the sea. Dec 04 '14 at 12:49
  • Comment as much as you like. I'm not sure I understand your question. In the notation used here, are you asking how to find $I$? – Git Gud Dec 04 '14 at 12:59
  • @GitGud, Yes. Imagine $f:I\rightarrow \mathbb{R}$. then, $fog:g^{-1}(I)\rightarrow \mathbb{R}$, no? If function $g$ is not invertible on that interval, then how would we be sure that we're finding the correct interval in the domain of $g$? – An old man in the sea. Dec 04 '14 at 13:11
  • The interval $I$ has to be given for the problem to make sense. It's usually implied or (as in most cases) irrelevant (in the sense that it doesn't really matter for the purposes of practicing finding antiderivatives). A question could ask to find a maximal interval I. To find it simply find an antiderivative by the regular means (even if the process is ultimately incorrect since you don't know what $I$ is), then check the maximal domains of definition of the antiderivative you get to find candidates for $I$. I'm afraid the present comment is going to do more harm than good. – Git Gud Dec 04 '14 at 13:16
  • @GitGud Do you know where I can study more about this? Or (and?) you could write an answer explaining this in more detail :) – An old man in the sea. Dec 04 '14 at 13:21
  • I'm afraid I'm not sure quite sure I understand your question. I have just been saying stuff that may or may not help hoping for the best. The reason why I don't understand what you want is that I can't decide if you really want to know about $I$ or if you want to know about the substitution function. Regarding $I$, I don't think you'll find somewhere to read about this, it's something authors don't wish to waste paper on, for whatever reason. – Git Gud Dec 04 '14 at 13:33
  • @GitGud I'll have to be away for a few hours. If you decide to write an answer, could you please talk about both the substitution function, and the interval I? Either way, do you know of some reference for finding primitives by substitution? Integration by substitution the wiki ref. suits me fine.Thanks ;) – An old man in the sea. Dec 04 '14 at 13:42
  • Just letting you know that me not answering your question is not unwillingness to help, but rather not knowing exactly how. – Git Gud Dec 13 '14 at 12:17
  • @GitGud no problem ;) – An old man in the sea. Dec 14 '14 at 07:44

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