That is because the derivative of the (r^2 -x^2) is -2x dx

You have only x dx in the original integral, so, in order for you to differentiate the integrand, the x dx must become -2x dx.

So you multiply and divide at the same time the x dx by -2.

In the integration,

V = (-4pi/3)[(r^2 -x^2)^(3/2)]|(0 to r),

V = (-4pi/3)[(r^2 -r^2)^(3/2) -(r^2 -0)^(3/2)]

V = (-4pi/3)[(0) -(r^2)^(3/2)]

V = (-4pi/3)[-r^3]

And so the negatives disappeared again.