<p>Given <br>\(\int x^5 e^{-4x^3} dx = \frac{1}{48} e^{-4x^3} f(x) + C\)<br>Then \(f(x)\) is equal to:</p>
Step-by-Step Solution
Key Concept: Use substitution u = -4x³ to convert the integral into a standard form, then apply integration by parts to the resulting exponential expression.
<p><strong>Step 1:</strong> Let u = -4x³, then du = -12x² dx, so x² dx = -du/12</p><p>Rewrite: ∫x⁵e^(-4x³) dx = ∫x³ · x² · e^(-4x³) dx = ∫(u/(-4)) · e^u · (-du/12)</p><p><strong>Step 2:</strong> Simplify: = (1/48)∫u·e^u du</p><p><strong>Step 3:</strong> Apply integration by parts to ∫u·e^u du with v = u, dw = e^u du:</p><p>∫u·e^u du = u·e^u - ∫e^u du = u·e^u - e^u = e^u(u - 1)</p><p><strong>Step 4:</strong> Therefore: ∫x⁵e^(-4x³) dx = (1/48)e^(-4x³)(-4x³ - 1) + C</p><p><strong>Step 5:</strong> Comparing with (1/48)e^(-4x³)f(x) + C:</p><p>∴ <strong>f(x) = -4x³ - 1</strong></p>
Correct Answer: A