Why the integral of 1/x requires an absolute value
In short: In calculus, integration is often viewed as the inverse of differentiation. This video explains why finding the antiderivative of 1/x requires a slight but crucial adjustment to account for negative numbers.
While the derivative of ln(x) is known to be 1/x, students must consider the domain when integrating. Because ln(x) is undefined for negative values of x, the antiderivative of 1/x cannot simply be ln(x).
To ensure the function remains valid across all non-zero real numbers, the correct antiderivative is expressed as ln(|x|). This distinction ensures that the derivative of the resulting function consistently yields 1/x regardless of whether x is positive or negative.
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Sources
- Summary by Intellect · source: Indefinite integral of 1/x | AP Calculus AB | Khan Academy
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