Consider the integral

$$I(x)=\int_{0}^{1} \frac{1}{\sqrt{t(1-t)}} \exp [i x f(t)] d t$$

for real $x>0$, where $f(t)=t^{2}+t$. Find and sketch, in the complex $t$-plane, the paths of steepest descent through the endpoints $t=0$ and $t=1$ and through any saddle point(s). Obtain the leading order term in the asymptotic expansion of $I(x)$ for large positive $x$. What is the order of the next term in the expansion? Justify your answer.