Let \(\large \displaystyle \dfrac{{d}}{dx } f(x) = \dfrac{e^{\sin x}}{x} \) where \( x > 0.\)

If \(\large \displaystyle \int_1^4 \dfrac{3}{x} e^{\sin x^3} \, dx = f(k) - f(1).\)

Find the value of \(k.\)

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