Assume I have an integral $$\int_{t_{1}}^{t_{2}} H(t)F(t) dt$$ where $H$ and $F$ are smooth and $|t_{1}-t_{2}$| is small. Assume the indefinite integral $$\int H(t)F(t)dt = G(t)$$ for some $G(t)$. Further, assume that $F(t_{1})=0$, $F'(t_{1}) \neq 0$.
I want to obtain an asymptotic formula for $\int_{t_{1}}^{t_{2}} H(t)F(t) dt$ using that $t_{1} \simeq t_{2}$. Let $' = \frac{d}{dt}$. Then we have
$G'= HF$ and $G''(t) = H'F + HF'$ and then by expanding $G$ near $t = t_{1}$ we have $$G(t) = G(t_{1}) + (t-t_{1})G'(t_{1}) + \frac{(t-t_{1})^{2}}{2}G''(t_{1}) + \mbox{h.o.t.}$$ and plugging $t = t_{2}$, we obtain
$$\int_{t_{1}}^{t_{2}} H(t)F(t) dt = \frac{(t_{2}-t_{2})^{2}}{2}G''(t_{1}) + O(t_{2}-t_{1})^{3} \simeq \frac{(t_{2}-t_{1})^{2}}{2}H(t_{1})F'(t_{1})$$
What I want to know, under what general conditions on $F$ and $F'$ we can approximate asymptotically the last formula and write:
$$\frac{(t_{2}-t_{1})^{2}}{2}H(t_{1})F'(t_{1}) \sim \frac{(t_{2}-t_{1})^{2}}{2}H(t_{1})$$
Any ideas? Thank you!