Use the substitution to evaluate
The problem before this was using the same substitution, but to evaluate instead. I was able to do that one easily, but I don't see how I can use this substitution for this problem.
Observe that $\displaystyle u={\color{blue}1-x^2}\implies {\color{red}x^2}=1-u$, then $\displaystyle \,du={\color{green}-2x\,dx}$. The limits of integration change as well: $\displaystyle u(0)=1$ and $\displaystyle u(1)=0$
So $\displaystyle \int_0^1 x^3\sqrt{1-x^2}\,dx=-\tfrac{1}{2}\int_0^1{\color{green}-2x}\cdot {\color{red}x^2}\sqrt{{\color{blue}1-x^2}}{\color{green}\,dx}$ $\displaystyle \xrightarrow{u=1-x^2}{}-\tfrac{1}{2}\int_1^0\left(1-u\right)\sqrt{u}\,du=\tfrac{1}{2}\int_0^1u^{1/2}-u^{3/2}\,du$
Can you continue from here?
Good question.
You have the substitution $\displaystyle u=1-x^2$
so $\displaystyle \sqrt{1-x^2}=\sqrt{u}$
The problem here is: you want to find $\displaystyle x^3$ and $\displaystyle dx$ in terms of $\displaystyle u$.
Its easy; solve the substitution for $\displaystyle x$ then differentiate both side with respect to $\displaystyle x$ to get the $\displaystyle dx$.
Also find $\displaystyle x^3$ from your substitution.
Can you do that?