Hi there,

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 $\displaystyle \phi(t)$, in place of $\displaystyle x$ in the original function that we are willing to integrate. The only thing we have to be concerned about is to make sure that $\displaystyle \phi(t)$ takes as values all and only those real numbers which $\displaystyle x$ 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?