I understand that the derivative of $\ln(x)$ is equal to $\dfrac{1}{x}$. What I don't get is why the antiderivative of $1/x$ is $\ln(|x|)$ instead of $\ln(x)$. Why is the absolute value sign necessary? I would like a simple explanation as I am new to calculus. Thanks!
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Go to the addendum of this post. Let's continue from the work there by letting $f(x)=x$ then integrating the equation in each case. Then, we have that $\int\frac1x,\mathrm dx$ equals $\ln|x|+C_1$for negative $x,$ and equals $\ln|x|+C_2$for positive $x.$ (Notice that this integral exists if and only if the domain of integration is a subset of $(-\infty,0)$ or a subset of $(0,\infty).$ – ryang Jul 29 '22 at 06:10
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$\ln (x)$ is only defined on $(0,\infty)$, so you need to ensure it's accounted for as an antiderivative of $\frac{1}{x}$, since $\frac{1}{x}$ is defined over all values except $0$.
For $\frac{d}{dx} \ln (x)$ we are implicitly restricting the domain to $(0, \infty)$.
Evan Tseng
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Annika
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