<p>If \(\int x^5 e^{-x^2} dx = g(x)e^{-x^2} + C\), where \(C\) is a constant of integration, then \(g(-1)\) is equal to:</p>
Step-by-Step Solution
Key Concept: Use substitution u = -x² to convert the integral into a polynomial times e^u form, then apply integration by parts strategically on the resulting polynomial term to find g(x).
<p><strong>Step 1:</strong> Use substitution u = -x², so du = -2x dx, giving x dx = -du/2. Rewrite x⁵e^(-x²) = x⁴ · x · e^(-x²) = (u²) · e^u · (-du/2)</p><p><strong>Step 2:</strong> ∫x⁵e^(-x²) dx = -½∫u² e^u du. Apply integration by parts twice on ∫u²e^u du:</p><p>Let v = u², dw = e^u du → ∫u²e^u du = u²e^u - 2∫u e^u du</p><p><strong>Step 3:</strong> For ∫u e^u du: Let v = u, dw = e^u du → ∫u e^u du = ue^u - ∫e^u du = ue^u - e^u = e^u(u - 1)</p><p><strong>Step 4:</strong> ∫u²e^u du = u²e^u - 2e^u(u - 1) = e^u(u² - 2u + 2)</p><p><strong>Step 5:</strong> ∫x⁵e^(-x²) dx = -½e^u(u² - 2u + 2) + C = -½e^(-x²)(x⁴ + 2x² + 2) + C</p><p><strong>Step 6:</strong> Comparing with g(x)e^(-x²) + C, we have g(x) = -½(x⁴ + 2x² + 2)</p><p><strong>Step 7:</strong> g(-1) = -½(1 + 2 + 2) = -½(5) = -5/2</p><p>∴ Answer: D</p>
Correct Answer: D