What is a necessary condition for the Fundamental Theorem of Calculus Part 1 to apply?

Evaluate the derivative of \( \int_{1}^{x^3} \sin(t) \, dt \) with respect to x.

Use the Fundamental Theorem of Calculus to compute ${\int }_2^4(x^2-4x+5)dx$.

Evaluate ${\lim }_{x\to 3}\frac{{\int }_3^x\sqrt{t^2-2t+6}dt}{x^2-9}$.

Let $y={\int }_{x^3}^{2x^3}{e}^{t}dt$. Find $\frac{dy}{dx}$.