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Let's use u-substitution for the integral. Let u = x³.Then, du/dx = 3x², so du = 3x² dx, or x² dx = (1/3) du.Change the limits of integration:When x = 0, u = 0³ = 0.When x = 2, u = 2³ = 8.Substitute these into the integral:∫₀⁸ k * e^u * (1/3) du = (k/3) ∫₀⁸ e^u du.Integrate e^u: (k/3) [e^u]₀⁸.Evaluate at the limits: (k/3) (e⁸ - e⁰) = (k/3) (e⁸ - 1).We are given that this integral equals (2/3) (e⁸ - 1).So, (k/3) (e⁸ - 1) = (2/3) (e⁸ - 1).Comparing both sides, we can see that k/3 must equal 2/3.Therefore, k = 2.