From Bruce van Brunts's book the derivation for the Euler-Lagrange equation uses the Taylor's theorem in the following way: Given $\hat{y}=y+\epsilon\eta$,
$J(\hat{y})-J(y) = \int_{x_0}^{x_1} f(x,\hat{y},\hat{y}')dx - \int_{x_0}^{x_1} f(x,y,y')dx = \int_{x_0}^{x_1} \left\{ f(x,y,y') + \epsilon \left\{ \eta\frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'}\right\} +O(\epsilon^2) - f(x,y,y')\right\}dx =\epsilon\int_{x_0}^{x_1}\left( \eta\frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'} \right)dx + O(\epsilon^2)$
My question is: why the term $O(\epsilon^2)$ came out of the integral in the last step? From Taylor's formula, this term should be a function of $x$: $r(0,\epsilon\eta(x),\epsilon\eta'(x))$ that satisfies $\lim \limits_{v \to 0} \frac{r(v)}{||v||}=0$. I really did not understand that step.