I've had this very basic question on the back of my mind for 2 almost years now and I think I've found a satisfactory answer. The question is this most simple one: how do we justify a change of variables such as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int e^{ax}dx = \int \frac{1}{a}e^udu[/tex]

in an indefinite integral? My "solution" is this... We are looking for a primitive [itex]F(\beta)[/itex] to [itex]f(\beta)[/itex]. By one version of the fondamental thm, as soon as f is continuous, such a primitive is given by

[tex]F(\beta)=\int_{\phi(\alpha)}^{\phi(\beta)}f(x)dx = \int_{\alpha}^{\beta}(f\circ \phi)(s)\frac{d\phi}{ds}ds[/tex]

where [itex]\phi:(\alpha,\beta)\rightarrow (\phi(\alpha), \phi(\beta))[/itex] is [itex]C^1[/itex]. (Ths is just the change of variable thm)

This may be total nonsense, but my eyes are itching from tiredness so I wouldn't know. Any comments on the above or an opinion on how to justify change of variable in indefinite integrals will be warmly appreciate tomorrow morning.

Good night!

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# Change of variable in indefinite integrals

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