Your calculus prof lied to you (probably)
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What is the indefinite integral of 1/x? The most common answer is ln|x|+C. This is a more satisfying answer than just ln(x)+C without absolute values because the domain of ln|x| and 1/x match (all real numbers except zero). However, this still isn't quite right as the indefinite integral is defined as ALL anti-derivatives. The slightly pedantic full answer is all piecewise defined functions that look like ln(x)+C for x bigger than 0 and ln(-x)+D for x smaller than 0, that is having different constants on both sides of the hole in the domain. This still satisfies a corollary of the Mean Value Theorem that asserts any two anti-derivatives differ by at most a constant provided we are on an open interval, which is what happens on each side of x=0. The animations in this video were made with Geogebra.
0:00 The indefinite integral of 1/x
0:40 Why is ln|x| a better answer than ln(x)?
3:14 Why do we always add +C ?
3:58 What really is an indefinite integral?
5:03The full answer
7:17 The Mean Value Theorem and the +C
8:30 Check out Brilliant.org/TreforBazett
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