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?



Asymptotic estimation of an integral
(This question should be on math.stackexchage.com.)


Substitute $v=t^{-1/2}u$, then it becomes
$$ t^{-1/2} \int\_0^{t^{1/2}} e^{-2u^2}\bigl(1 + O(u^3/t^{1/2})\bigr)\,du
= \sqrt{\frac{\pi}{8t}} + O(t^{-1}).$$


