Let $\frac{d}{dx} \text{P}(x)=\frac{e^{\sin x}}{x}, x > 0$. If $\int ^{2}_{1}\frac{3}{x} e^{\sin x^{3}} dx= \text{P}(k) - \text{P}(1)$, then which of the following is a possible value of $k$?
$2$
$4$
$8$
$16$
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