Undergrad Confusion on a simple integration

[rude man]

TL;DR

Confused on integrating $\int dx/(x^2 + a^2)^(1/2)$

The tables and Wolfram Alpha say
$\int dx/(x^2 + a^2) ^{1/2} = log~ [( x^2 + a^2)^{1/2} + x]$.

So if a=0 we get as answer
$log (x + x) = log( 2x)$ if $(x^2)^{1/2} = x, or$ log (x - x) = log( 0) $if$ (x^2)^{1/2} = -x

but surely $\int dx/(x^2)^0.5 = \int dx/x = log x$ ?

Only 2 roots of $x^2$ right?

 

[fresh_42]

My list of integrals (Wikipedia) allows this formula only for $a>0$.

 

[phyzguy]

log(2x) = log(2) + log(x), so log(2x) and log(x) differ only by a constant of integration.

 

[rude man]

log(2x) = log(2) + log(x), so log(2x) and log(x) differ only by a constant of integration.

Thank you a lot phyzguy! Could have driven me batty who knows for how long!