Find the definite integral for: \begin{equation} \int_C x^2y\,ds ,\quad C: \begin{cases} x = \sin(t^3-4t) \\ y = \cos(3t^2) \end{cases} ,\quad t\in\left[1,2\right]. \end{equation}
Normally I'd substitute: \begin{equation} ds = \sqrt{x'(t)^2 + y'(t)^2}dt. \end{equation}
But in this case the parameterized curve has polynomials inside the trigonometric function which makes it messy. I guess maybe the whole idea is to substitute somehow those polynomials?