Do you consider the following to be true or false and why? ⇒

Question

Do you consider the following to be true or false and why?

dx/x=-dx/-x ⇒ ln x= ln(-x)

Answer 1

The integral of 1/x dx is the absolute value of the logarithm of x. It will be a mistake saying that the integral of 1/x dx is log(x). The reason of the presence of the absolute value is exactly to avoid the argument of the logarithm to be negative.

Hence, the correct sequence of equalities will be dx/x = -dx/-x ⇒ ln |x| = ln |-x| and in this case you will not have problems related to x being negative.

Best,

Davide