Hi guys, I'm stuck on something trivial.

View attachment 42843 letting x = (u-4)^2

I get to the correct solution in terms of u: 2u - 8ln|u| + c

However, my confusion begins when changing the answer back to be in terms of x. Since x = (u-4)^2 , u = x^(1/2)+4 OR u =-x^(1/2)+4

So shouldn't there be two solutions in terms of x? One for the positive case and one for the negative case? Solutions across the internet and in back of book give the positive answer only. Could someone please elaborate and clear this up for me.

maybe think of it as letting sqrt(x) = u -4,

so then 1/2sqrt(x) dx= du

so dx = 2sqrt(x)du = 2(u-4)du

from here u get the same result but now the substitution back in makes sense as u = 4+sqrt(x)

so this is why the positive square root is the typical answer

alternatively, if u = 4-sqrt(x), then sqrt(x) = 4-u, if u sub this in then after some manipulation u get that the integral is -2u -8ln(8-u) +C

which becomes -2(4-sqrt(x))-8ln(4+sqrt(x)) +C = 2sqrt(x) - 8ln(4+sqrt(x)) +C after absorbing stuff into the constant

which is completly equivalent to when u do the u-sub as u = 4+sqrt(x)

so actually it does not matter, so long as u pick which sign at the beginning, though obviously the sub u = 4+sqrt(x) gives an easier result

the issue comes when u cross out sqrt((u-4)^2), this actually becomes |u-4|, in general you pick the positive case by "cancelling" it out without thinking about it but you then need to consider the positive case of the substitution only. this is why it's a better idea to just sub sqrt(x) = u-4, it removes the confusion with the absolute value