Question #67206

Let $x(t) = \int_0^t \frac{\sin u}{u} du$ and $y(t) = \int_0^t \frac{\sin^2 u}{u^2} du$ for $t \in (-\pi, \pi) \setminus \{0\}$. Let $M(t) = \begin{pmatrix} \frac{dy}{dx} & \frac{1}{t}\frac{d^2y}{dx^2} + \frac{1}{t}\int_0^t e^{u^2} du \\ \frac{1}{t}\frac{d^2y}{dx^2} - \frac{1}{t}\int_0^t e^{u^2} du & \frac{d^3y}{dx^3} \end{pmatrix}$. For $t \neq 0$, $M(t)$ can be uniquely expressed as $S(t) + K(t)$, where $S(t)$ is a symmetric matrix and $K(t)$ is a skew-symmetric matrix. If $A = \lim_{t \to 0} \det(S(t))$ and $B = \lim_{t \to 0} \det(K(t))$, find the value of $18A + B$.