Problem about substitution in integration

[kelsiu]

It is common that we replace \(\displaystyle \int u(x)v'(x)dx\) by \(\displaystyle \int udv\) where both u and v are continuous functions of x. My question is, must we ensure that u can be written as a function of v before applying this? The above substitution method is involved in the proof of integration by parts but I cannot find textbooks that addressed this point.

I find it easier to understand for definite integrals because I can think it as summation. But for indefinite integrals it is just anti-derivative and I am not sure how this kind of replacement is valid.

 

[stapel]

It is common that we replace \(\displaystyle \int u(x)v'(x)dx\) by \(\displaystyle \int udv\) where both u and v are continuous functions of x.

Um... what is the basis for this claim?