<p>Let \(\dfrac{d}{dx}F(x) = \left(\dfrac{e^{\sin x}}{x}\right)\), \(x > 0\). If \(\displaystyle\int_1^4 \dfrac{3}{x} e^{\sin x^3} dx = F(k) - F(1)\), then one of the possible value of \(k\) is</p>
Step-by-Step Solution
Key Concept: Use substitution u = x³ to transform the integral into the form of F(u), then match coefficients to find k. The key is recognizing that the integrand structure matches the derivative definition after appropriate substitution.
<p><strong>Step 1:</strong> Given that dF/dx = e^(sin x)/x, we need to evaluate ∫₁⁴ (3/x)e^(sin x³) dx</p><p><strong>Step 2:</strong> Use substitution u = x³, so du = 3x² dx, which means dx = du/(3x²)</p><p><strong>Step 3:</strong> Rewrite the integral: ∫₁⁴ (3/x)e^(sin x³) dx = ∫₁⁴ (3/x)e^(sin u) · (du)/(3x²) is incorrect. Instead, directly substitute: when x = 1, u = 1; when x = 4, u = 64</p><p><strong>Step 4:</strong> Note that 3x² dx = du, so (3/x) dx = du/x². Actually, let's reconsider: ∫₁⁴ (3/x)e^(sin x³) dx. Let u = x³, then 3x² dx = du, so dx = du/(3x²). The integral becomes ∫₁⁶⁴ (3/x) · e^(sin u) · du/(3x²) = ∫₁⁶⁴ e^(sin u)/(x³) du. Since x³ = u, this is ∫₁⁶⁴ e^(sin u)/u du = F(64) - F(1)</p><p><strong>Step 5:</strong> Comparing with F(k) - F(1), we get k = 64</p><p>∴ Answer: D (k = 64)</p>
Correct Answer: D