To solve the integral, we will first simplify the expression involving the Chebyshev polynomial of the second kind, $U_n(x)$. Step 1: Simplify the argument of $U_n$. Let $x = a \tan \theta$. Then $dx = a \sec^2 \theta \, d\theta$. When $x=0$, $\theta=0$. When $x \to \infty$, $\theta \to \pi/2$. The term $(a^2 + x^2)^{1/2}$ becomes $(a^2 + a^2 \tan^2 \theta)^{1/2} = (a^2 \sec^2 \theta)^{1/2} = a \sec \theta$. The argument of $U_