What is the integration of 1/x?

1 Answer
Jim H
Apr 11, 2015

int 1/x dx = ln abs x +C

The reason depends on which definition of ln x you have used.

I prefer:
Definition: lnx = int_1^x 1/t dt for x>0

By the Fundamental Theorem of Calculus, we get: d/(dx)(lnx) = 1/x for x>0

From that and the chain rule, we also get d/(dx)(ln(-x)) = 1/x for x<0

On an interval that excludes 0, the antiderivative of 1/x is
lnx if the interval consists of positive numbers and it is ln(-x) if the interval consists of negative numbers.

ln abs x covers both cases.

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