$($a$)$ Prove that ${\displaystyle \underset{0}{\overset{}{}}}\frac{1}{t^2+x}dt$ converges uniformly on $x1.$

$($b$)$ Prove that ${\displaystyle \underset{0}{\overset{}{}}}\frac{1}{t^2+x}dt=\frac{}{2\sqrt[\phantom{2}]{x}}$ for $x1.$

$($c$)$ Use $($b$)$ to evaluate ${\displaystyle \underset{0}{\overset{}{}}}\frac{1}{(x^2+1{)}^2}dx.$