I am confused as to what dx truly is. I am doing some u-substitution problems and this is what I came across:
∫2x(x−1)1/2dx
u=x−1 and therefore du=1
when we substitute we get:
2∫(u3/2+u1/2)du
(here the du simply replaces the dx because our variable changed)
In another example:
∫4x5(x2+1)1/3dx
u=x3+1
du=3x2
therefore it becomes:
∫4(u−1)(du/3)(u)1/3
-here my teacher didn't put dx at the end, she just left it off
So my question is this: why is it that sometimes dx and du are treated as values that can be multiplied to other terms in the integrand and sometimes they are simply treated as a command (do "blank" with respect to x, or u or whatever is used)?
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