Can anyone explain to me please why is this solution to this integral is wrong? So this is how I initially wanted to find this antiderivative:
$I_1=\mathbf{}\int\frac{x^3}{\sqrt{4-x^4}}dx,x\in(0,1)$
I substituted $\ x^4=u, d(x^4)=d(u)\Leftrightarrow 4x^3dx=du\Leftrightarrow x^3dx=\frac{du}{4}$
to get $I_2=\frac{1}{4}\cdot\int\frac{1}{\sqrt{2^2-u}}du$ and I rewrote it as $I_2=\frac{1}{4}\cdot\int\frac{1}{\sqrt{2^2-(\sqrt{u})^2}}du$ in an attempt for me to use the formula
$\int\frac{1}{\sqrt{a^2-x^2}}dx=\arcsin(\frac{x}{a})$ to get $I_2=\frac{1}{4}\arcsin(\frac{\sqrt{u}}{2})$
and $I_1=\frac{1}{4}\arcsin(\frac{\sqrt{x^4}}{2})=\frac{1}{4}\arcsin(\frac{x^2}{2})$
Now I know that by substituting $4-x^4=u$ you will get the right antiverivative $-\frac{\sqrt{4-x^4}}{2}$ which I verified by taking it's derivative and my first substitution is wrong but don't understand why. I graphed them on desmos and they look similar to some extent.

I asked my high school math teacher why it is wrong and she told me that it is not $(\sqrt{x})^2$ in the formula I tried to use but $x^2$ and so the formula was used wrong and that is why the solution is wrong but that didn't clarify my question too much. Can anyone explain to me why writing $u=(\sqrt{u})^2$ and using the formula was wrong? Why does using $(\sqrt{u})^2$ instead of just $u^2$ is wrong?
 A: After your edits, your initial substitution is correct, though I would have chosen to let $u=4-x^4$.  Still, what you have obtained is doable.  You have
$$
\frac{1}{4} \int \frac{1}{\sqrt{4-u}} \ du.
$$
This can be done either by eyesight or with the substitution $w=4-u$.  This will reduce to an integral involving $\int \frac{1}{\sqrt{w}} \ dw$ which is easy with the power rule.
Your arcsine attempt is wrong because you have not accounted for the chain rule.  It is true that
$$
\int \frac{1}{\sqrt{1-t^2}} \ dt = \arcsin(t)+C
$$
but that is not what you have done.  You have let $t = \sqrt{u}$ (roughly, let's ignore the $4$ for now) but you never "corrected" for the $dt$ effect. In other words, you substituted in the $\arcsin$ integral without ever handling the differential part of the substitution.
Sometimes a different example with the same mistake is illuminating.  It is a basic fact that
$$
\int e^t \ dt = e^t + C.
$$
Can we just replace $t=\sqrt{x}$ and get
$$
\int e^{\sqrt{x}} \ dx = e^{\sqrt{x}} + C?
$$
No, this is false which you can check by differentiating.
Or, do the same thing with a more obviously wrong example.  We know that
$$
\int t \ dt = \frac{t^2}{2}+C.
$$
Now let $t=\sqrt{x}$ and conclude (falsely) that
$$
\int \sqrt{x} \ dx = \frac{x}{2}+C,
$$
which is clearly wrong.  The problem?  I never handled the $dt$ part.
