Solve the integral question

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_