$\displaystyle \int x^2\cos(x^3)   dx =$

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Solve using $u$-substitution.

$$u = x^3$$ $$du = 3x^2   dx$$ $$dx = \frac{du}{3x^2}$$ $$\int x^2\cos(\colorbox{aqua}{$x^3$})   \colorbox{yellow}{$dx$} = \int x^2\cos{\colorbox{aqua}{$(u)$}}\colorbox{yellow}{$\dfrac{du}{3x^2}$}$$ $$= \int \frac{1}{3}\cos(u)   du$$ $$= \frac{1}{3}\sin(u) + C$$ $$= \boxed{\frac{1}{3}\sin(x^3)+C}$$

Though a bit more tedious, feel free to take the derivative of each answer choice.