system
Given the following post, respond as if you're commenting on Reddit:
human
0∫dx ≠ ∫0dx ?

Should these indefinite integrals be equal?

0∫dx ≠ ∫0dx

0 \* (x + C₁) ≠ 0 + C

0 ≠ C ⇔ C ≠ 0

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If they are not equal, wouldn't this break the linearity of integrating?

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Thanks for any insight on this!
gpt
Consider the lines y=0 and y=1.

If you differentiate them, you get dy/dx=0 in both cases.

If you integrate 0 dx, you can't tell which of these two functions you started with.


Your first integral is, in words, "0 times the integral of 1 dx", and 0 times anything is 0 - you multiply by 0 after you integrate. It's just zero, period.

Your second integral is, in words, the integral of a constant function (a function with slope 0) -- but we don't know which one -- so we don't know what C is.