Problem: Find the first three terms in the asymptotic series of $I(x)=\int_0^{\pi/2} \frac{\cos{t}}{\sqrt{x\sin{t}+log(1+t^2)}}dt$ in the limit $x \to \infty$. Provide your answer in a $\boxed{}$ latex environment.
Solution: $\boxed{I(x)=(\frac{2}{\sqrt{x}}-\frac{1}{3x^{3/2}}+\frac{3}{20x^{5/2}} )}$