Inspired limits problem

Calculus Level 3

\[ \mathscr{E} = \displaystyle \lim_{x \to 0} \dfrac{\displaystyle \int_{x^2}^{x^3} \sin t^3 \, dt}{x^4} \]

Find \( \left \lfloor 1000 \mathscr{E} \right \rfloor \).

\[\] Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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