In my textbook "Mathematical analysis I" we saw something called "Boundary to boundary transformation of an integral" (Note that my textbook is a Dutch textbook, I've tried to translate the name the best I could, but can't find much about it on the internet so i don't know if it's a correct translation).

Suppose $a<b$, $\theta \in C^1[a, b]$ and $f$ continuous over the interval formed by the image $\theta [a,b]$, then:

\begin{equation} \int\limits_a^b f(\theta(x))\theta'(x)dx = \int\limits_{\theta(a)}^{\theta(b)} f(y)dy \end{equation}

The $C^1[a,b]$ used in the theorem is defined in the textbook as:

A function $f$ is of class $C^1$ over an interval $[a,b]\subseteq\mathcal{D}_f$ if the following 3 properties are true:

  • $f$ is continuous over $[a,b]$
  • $f'$ exists and is continuous over $]a,b[$
  • $f'$ has a right hand limit in $a$ and a left hand limit in $b$

So the question now is what this actually means and what actually happens, because I don't quite get it. My best guess is that it has something to do with transforming an integral seen from the x-axis, to an integral seen from the y-axis, while keeping the same result. But I don't really see how this happens, especially int the left integral. What does the $f(\theta(x))$ and $\theta'(x)$ do and where do they come from. I also do not know what this $\theta$ function is. I also include a picture with a drawing underneath.

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    $\begingroup$ The English terminology for this is “$u$-substitution”. $\endgroup$ – Clayton Dec 26 '18 at 19:25
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    $\begingroup$ Is this really u-substitution? Well i do know what u-substitution is, so i do have an idea of what this resembles, it already makes a lot more sense now, I will take a look at it again with this in mind! I guess i was confused by the weird name for it. $\endgroup$ – Viktor Dec 26 '18 at 19:27
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    $\begingroup$ Certainly; take $u=\theta(x)$ then $du=\theta’(x)\,dx$ and the new bounds will be $\theta(a),\theta(b)$. $\endgroup$ – Clayton Dec 26 '18 at 19:30
  • $\begingroup$ It all makes sense now, i was just looking at it all wrong. Thanks for your help! $\endgroup$ – Viktor Dec 26 '18 at 19:31
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    $\begingroup$ This is also generally called the change of variables theorem. $\endgroup$ – Sean Roberson Dec 26 '18 at 19:47

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