$(a)$ Prove the reduction formula

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

$(b)$ Apply $(a)to$ find

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^2}{(a^2+x^2{)}^2}dx.$