do you have an example? it makes things clearer.
I'm currently trying to grasp the idea standing behind Euler's substitutions in integration.
Please correct me if I miss the point entirely - when integrating by substitution we simply put a function, let's say , in place of in the original function that we are willing to integrate. The only thing we have to be concerned about is to make sure that takes as values all and only those real numbers which may take as values as well.
Is that right?
Knowing that, what is it that makes us so sure that Euler's substitutions will work exactly the way described?
The thing is, I've been reading Springer Online Reference Works and was trying to make sense out of the last paragraph (starting with the words "Geometrically, the Euler substitutions mean that...").
The way I understand it is this.
Suppose we take any point such that . Now, let's take the straight line . We may stipulate the value of as we please. This proves... Well, I'm not sure what exactly does it prove. And that's basically where I'm stuck.