This integral can be simplified to $$ \begin{equation*} \int_{x_0}^{x_0+h}f(x)dx\approx \int_{x_0}^{x_0+h}\left(\frac{x-x_0}{h}f(x_0+h)-\frac{x-(x_0+h)}{h}f(x_0)\right)dx, \end{equation*} $$ resulting in $$ \begin{equation*} \int_{x_0}^{x_0+h}f(x)dx=\frac{h}{2}\left(f(x_0+h) + f(x_0)\right)+O(h^3). \end{equation*} $$ Here we added the error made in approximating our integral with a polynomial of degree \( 1 \).