why is the following U-Substitution wrong?

Question

It is known that

$$ \int \frac{1}{\sqrt{1-x^2}} dx = arcsin(x)+c$$

this can be done utilizing u-substitution $ x = sin(u) $

However,

i can let $ u = 1-x^2 $

$dx = -2u \, du $

which gives the integral

$$ -\int \frac{2u}{\sqrt{u}} du $$

which then returns $$ \frac{-4\sqrt{u^3}}{3} $$

which is clearly wrong.

So why is this wrong?

thanks very much for your help and apologies about the elementary question.

REMARK

Upon checking other posts , a common problem has to do with the range of the function being substituted in.

Example

What is wrong with the following u-substitution?

Answer 1

Your alternative method is incorrect, because if $u=1-x^2$ then $du=-2x\,dx$.

You wrote $dx=-2u\,du$; that is different.