Let $h(t)=i\left(t+t^{2}\right)$. Sketch the path of $\operatorname{Im}(h(t))=$ const. through the point $t=0$, and the path of $\operatorname{Im}(h(t))=$ const. through the point $t=1$.

By integrating along these paths, show that as $\lambda \rightarrow \infty$

$$\int_{0}^{1} t^{-1 / 2} e^{i \lambda\left(t+t^{2}\right)} d t \sim \frac{c_{1}}{\lambda^{1 / 2}}+\frac{c_{2} e^{2 i \lambda}}{\lambda},$$

where the constants $c_{1}$ and $c_{2}$ are to be computed.