Integration by substitution using implicit function relation

Question

When trying to find the integral

$$I = \int (x\sqrt{2x + 5}) \mathrm{d}x$$

if we use the implicit substitution $u^2 = 2x + 5$, substituting into the integrand gives the sub-expression $\sqrt{u^2}$. We usually simplify it to $+u$ and proceed smoothly. My question is, why don't we consider $-u$ here?

EDIT:

I removed the explicit substitution, $u = 2x + 5$, let's focus on the implicit one.

Answer 1

By definition $\sqrt{x}$ indicates the positive (or principal) square root of $x$. If we want indicate all the two root we have to write explicitly $\pm\sqrt{x}$.

Answer 2

1. The implicit substitution $h(u) = 2x + 5,$ where $u$ is the new variable, is valid for each invertible function $h$.

2. In this case, since $h(u)=u^2,$ it is tacit that $h$ has domain $[0,\infty).$

   This is why $\sqrt{u^2}\big(=|u|\big)$ can be simply expressed as $u.$

3. Alternatively, we might specify $h$'s domain as $(-\infty,0],$ in which case $\sqrt{u^2}=-u$.