$\frac{d}{d x} \int_{a}^{b} \sin x^{r} d x$

Step 1 ：Find the derivative of the integral of sin(x^r) with respect to x using the Fundamental Theorem of Calculus: f(x) = sin(x^r)

Step 2 ：Calculate the derivative: \(r x^{r-1} \cos(x^r)\)

Step 3 ：Final Answer: \(\boxed{r x^{r-1} \cos(x^r)}\)