I wanted to look more closely at the change of variables technique for integrals (u-sub) since it showed up in a physics problem I was trying to solve. I feel like change of variables sort of shows the following definition of a differential $$du=f'\left(x\right)dx=\frac{du}{dx}dx$$ This identity was shown to me in Calc 1, and it is frequently used to perform u-sub. From what I know, if $u=f(x)$ in an integral, then $du=f’(x)dx$, and that can be substituted in. I recently looked at the proof for u substitution, and neither the definition of a differential, or a substitution, are really used in it though. if: $$\frac{d}{dx}F\left(φ\left(x\right)\right)=f\left(φ\left(x\right)\right)φ'\left(x\right)$$ Then: $$\int_{a}^{b}f\left(φ\left(x\right)\right)φ'\left(x\right)dx=\int_{a}^{b}\frac{d}{dx}F\left(φ\left(x\right)\right)dx$$ $$=F\left(φ\left(b\right)\right)-F\left(φ\left(a\right)\right)$$ $$=\int_{φ\left(a\right)}^{φ\left(b\right)}f\left(u\right)du$$ While this proof does ‘undo’ the chainrule, the substitution $u=φ(x)$ is never used, and neither is the definition of a differential. Thats odd to me since those tools are used to perform u sub. I thought u-sub would work in the following way: Given: $$\int_{a}^{b}f\left(φ\left(x\right)\right)φ'\left(x\right)dx=\int_{a}^{b}f\left(φ\left(x\right)\right)\frac{dφ}{dx}dx$$ If $u=φ\left(x\right)$, then $du=\frac{dφ}{dx}dx$ by the definition of a diffrential Substituting in: $$\int_{a}^{b}f\left(φ\left(x\right)\right)φ'\left(x\right)dx=\int_{φ\left(a\right)}^{φ\left(b\right)}f\left(u\right)du$$ I’ve always thought that if you change from $dx$ to $du$, then the bounds should now be in terms of $u$. Now that I think about it, I’m not sure why that's rigorously done. It seems more clearer with the first proof of change of variables, but that proof doesn’t use the definition of a differential. I guess I’m asking this because I was originally trying to derive the equation for change in kinetic energy from the following integral: $$m\int_{x_{1}}^{x_{2}}adx$$ I was given the following justification for rewriting the differentials: If $x=x(t)$, then $dx = \frac{dx}{dt}dt$ so: $$\frac{dv}{dt}dx = \frac{dv}{dt} \frac{dx}{dt} dt$$ $$=\left( \frac{dv}{dt} dt \right) \frac{dx}{dt}$$ Using the definition stated above, $dv=\frac{dv}{dt}dt$. Substituting in gives us: $$= \frac{dx}{dt} dv$$ Changing the bounds from x to v, we get:
$$m\int_{v\left(x_{1}\right)}^{v\left(x_{2}\right)}vdv\ =\ \frac{1}{2}mv\left(x_{2}\right)^{2}-\frac{1}{2}mv\left(x_{1}\right)^{2}=KE_f-KE_i$$ If I accept the definition of a differential this makes sense, but as I said I don’t really know why the bounds are changed. Am I misusing the definition of a differential that I assumed at the beginning? Is the definition of a differential really being used in change of variables? Or is it just a trick that's used to set it up? Thanks