I have some confusion about using Taylor Series to expand at L/2 arclength

[jiankang-star]

Thank your great work about ROAD_MODEL_FUSION, because it contains a lot of new methods about lane line fusion that worth to study.

But when I study the Taylor approximation, I have some confusion about using Taylor Series to expand at L/2 arclength.
The equation of $x(s)$ is below (page 19 in Road_Model_Fusion_version1.pdf)
$$x(s) = x_0 + \int_0^s cos(\varphi_0 + k_0 l + \frac{1}{2}\dot k l^2) dl$$
so the first term of Taylor Series should be
$$x(s_0) = x_0 + \int_0^{s_0} cos(\varphi_0 + k_0 l + \frac{1}{2}\dot k l^2) dl$$
and if let $s_0 = \frac{L}{2}$, the first term is
$$x(s_0) = x_0 + \int_0^{\frac{L}{2}} cos(\varphi_0 + k_0 l + \frac{1}{2}\dot k l^2) dl$$
and it can not be solved.
So, how to get the result $x(s) = x_0 + (...)s + (...)s^2 + (...)s^3 + (...)s^4$?

Looking forward to your reply!