Why the standard answer for the integral of 1/x is incomplete
In short: Most students are taught that the indefinite integral of 1/x is ln|x|+C. While this is a common simplification, it technically overlooks a fundamental rule of calculus regarding how anti-derivatives behave across gaps in a domain.
The video explores why the expression ln|x|+C is slightly inaccurate as a complete answer for the indefinite integral of 1/x. While ln|x|+C is often preferred over ln(x)+C because it shares the same domain (all real numbers except zero), it fails to account for every possible anti-derivative.
Technically, an indefinite integral must represent all anti-derivatives. Because the domain of 1/x has a break at x=0, the true answer consists of piecewise functions: ln(x)+C for values greater than zero and ln(-x)+D for values less than zero. This allows for different constants on either side of the gap, satisfying the Mean Value Theorem's corollary that anti-derivatives only differ by a constant within an open interval.
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Sources
- Summary by Intellect · source: Your calculus prof lied to you (probably)
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