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The velocity of a particle in the $xy$-plane is given by $v(t) = \left\langle t \sin {t^2} ,\ 3t^2 \cos {t^3} \right\rangle$.

If at $t = 0$, the particle is at the point $(3, -1)$, find the position of the particle at any time, $t$.

A

$\left\langle -\cfrac{\cos { t }^{ 2}}{2} +\cfrac{7}{2},\ \sin { { t }^{ 3 } } -1\right\rangle$

B

$\left\langle -\cfrac{\cos { t }^{ 2}}{2} +\cfrac{5}{2},\ \sin { { t }^{ 3 } } -1\right\rangle$

C

$\left\langle \cfrac{\cos { t }^{ 2}}{2} +\cfrac{5}{2},\ -\sin { { t }^{ 3 } } -1\right\rangle$

D

$\left\langle -\cos { { t }^{ 2 } } +4,\ \sin { { t }^{ 3 } } -1\right\rangle$

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