This seems impossible, but how do you do this integral [(1/x^2)(sqrt(1-x)/sqrt(x))]?

Answer 1

First, simplify:

∫1/x² * √(1-x)/√x dx
∫1/x² * √((1-x)/x) dx
∫1/x² * √(1/x-1) dx

Now, make a substitution. Let u=1/x-1, du=-1/x² dx

∫-√u du

Integrate:

-2u^(3/2)/3 + C

And resubstitute the original variable:

-2(1/x-1)^(3/2)/3 + C

Answer 2

It's much less hard than it looks. If you find the right simplification and substitution! ;-)

∫(1/x^2) . (√(1-x) / √(x)) dx
= ∫(1/x^2) (√(1/x - 1)) dx
Let u = 1/x - 1. Then du = (-1/x^2) dx, so we have
= ∫ (-√u) du
= -(u^(3/2)) / (3/2) + c
= (-2/3) (1/x - 1)^(3/2) + c.