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