What is the integral of 1x? Do you get ln(x) or ln|x| ?
In general, does integrating f′(x)/f(x) give ln(f(x)) or ln|f(x)| ?
Also, what is the derivative of |f(x)|? Is it f′(x) or |f′(x)|?
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What is the integral of 1x? Do you get ln(x) or ln|x| ?
In general, does integrating f′(x)/f(x) give ln(f(x)) or ln|f(x)| ?
Also, what is the derivative of |f(x)|? Is it f′(x) or |f′(x)|?
You have
∫1xdx=ln|x|+C
(Note that the “constant” C might take different values for positive or negative x
. It is really a locally constant function.)
In the same way,
∫f′(x)f(x)dx=ln|f(x)|+C
The last derivative is given by
ddx|f(x)|=sgn(f(x))f′(x)={f′(x)−f′(x) if f(x)>0 if f(x)<0