Question

What is the antiderivative of `1/(2x)`?

Answer 1

`1/2 ln abs x +C`

Explanation:

`d/dx(lnx) = 1/x` and `d/dx(ln(-x)) = 1/x`,

so the antiderivative of `1/x` is `ln abs x +C`

The constant `1/2` just hangs out in front.

Caution
In some treatments (James Stewart's Calculus , for example) there is a subtle difference between the antiderivative of a function with a discontinuity and the indefinite integral of such a function.

In such a treatment, the antiderivative of `1/x` is

`F(x) = {(lnx+C_1,"if",x < 0),(lnx+C_2,"if",x > 0) :}`

The integral is assumed to take place only on one side or the other, so `int 1/x dx = ln absx +C`