Positive/Negative after taking the square root

I'm not totally mathematically inept, but I find I'm getting confused
by the following:

I'm confused as to what happens to the +/- in the following examples:

As a simple example, integrate x^2

int{x^2 dx}

Let x^2 = u, then dx = du/(2x) and du = 2x, dx = +/- 2sqrt{u}

Then the integral becomes
=int{u/(+/-)2sqrt{u} du}
= +/- int{1/2sqrt{u} du}
= +/- 1/3u^1.5
= +/ 1/3x^3 (substitute back in for u)

This is obviously not correct, and the correct answer is in taking
only the positive square root when representing dx interms of u and
du. However, is there a mathematical/logical reason for discounting
the negative root? After all, if y^2 = x, then y = +/-sqrt{x}

A related problem I encountered when deriving the derivative of
arcsin:

y = arcsinx
x = siny

I have y' = 1/cosy, and wish to write y in terms of x

Now, sin^2y + cos^2y = 1

siny = x

thus x^2 + cos^2y = 1

cos^2y = 1-x^2

cosy = +/- sqrt{1-x^2}


Thus

y1 = +/- 1/sqrt{1-x^2}

Obviously, again this is incorrect. The correct answer is obtained by
discounting the negative root - again, what is the mathematical/
logical reason for this?
.