Exercise $2$: Wallis's Formulas

Evaluate the integrals

${\displaystyle \underset{-1}{\overset{1}{}}}(1-x^2)dx$ and ${\displaystyle \underset{-1}{\overset{1}{}}}(1-x^2{)}^{2}dx$

Use Wallis's Formulas to prove that

${\displaystyle \underset{-1}{\overset{1}{}}}(1-x^2{)}^{n}dx=\frac{2^{2n+1}(n!{)}^2}{(2n+1)!}$

for all positive integers $n.$