Is the antiderivative of 1/x: ln (x) or ln |x|
The second one. X has to be positive because it is the argument and the argument of a log or natural log must be positive
f the antiderivative of the 1/x were just ln(x) then you’d be saying that 1/x is not differentiable at any negative number. But, looking at the curve, we see that this part of the curve is just a reflection of the positive part across the origin. So since the positive part is differentiable, the negative part must be as well. Hence 1/x should be differentiable for negative numbers. Hence its antiderivative cannot be ln(x).
Note that there are some subtleties in what the set of antiderivatives of 1/x actually is because of the hole at x=0, but you’ll learn about that later. For now, just take ln|x| as the antiderivative.