I have an integral of the form $$ I = \int\limits^{1}_{0} \exp\left(\dfrac{vt}{(v+1)^2 + v^2} - vt\right) dv $$ and I want to prove that $I\leq c t^{-1}$ for the sufficiently large $t$, where $c$ is a constant independent of $t$. Can anyone give me some hints or references to prove this expansion?