<p><strong>Integer Answer</strong></p><p>If \(\displaystyle\int x^2 e^{3x}\,dx = e^{ax}\left[bx^2 - cx + d\right]+c\), then \(\dfrac{abc}{d}=\)</p>
Step-by-Step Solution
Key Concept: Use integration by parts twice on x²e^(3x), then match coefficients with the given form e^(ax)[bx² - cx + d] to identify a, b, c, d, and finally compute abc/d.
<p><strong>Step 1:</strong> Apply integration by parts to ∫x²e^(3x)dx with u = x², dv = e^(3x)dx</p><p>du = 2x dx, v = e^(3x)/3</p><p>∫x²e^(3x)dx = (x²/3)e^(3x) - (2/3)∫x·e^(3x)dx</p><p><strong>Step 2:</strong> Apply integration by parts again to ∫x·e^(3x)dx with u = x, dv = e^(3x)dx</p><p>du = dx, v = e^(3x)/3</p><p>∫x·e^(3x)dx = (x/3)e^(3x) - (1/3)∫e^(3x)dx = (x/3)e^(3x) - (1/9)e^(3x)</p><p><strong>Step 3:</strong> Substitute back:</p><p>∫x²e^(3x)dx = (x²/3)e^(3x) - (2/3)[(x/3)e^(3x) - (1/9)e^(3x)]</p><p>= (x²/3)e^(3x) - (2x/9)e^(3x) + (2/27)e^(3x)</p><p>= e^(3x)[x²/3 - 2x/9 + 2/27] + C</p><p><strong>Step 4:</strong> Factor out e^(3x) and match with e^(ax)[bx² - cx + d]:</p><p>Multiply by 9/9: e^(3x)[3x²/9 - 2x/9 + 2/27] = e^(3x)[(1/3)·(3x² - 2x + 2/3)]</p><p>Rewrite as: e^(3x)[3x² - 2x + 2/3]/9 → multiply inside bracket by 9: e^(3x)[9x² - 6x + 2/3]·(1/9)</p><p>Better approach: e^(3x)[x²/3 - 2x/9 + 2/27] = (e^(3x)/27)[9x² - 6x + 2]</p><p>Comparing with e^(ax)[bx² - cx + d]: a = 3, b = 9, c = 6, d = 2 (with factor 1/27)</p><p><strong>Step 5:</strong> Calculate abc/d = (3 × 9 × 6)/2 = 162/2 = 81</p><p>∴ Answer: <strong>81</strong></p>
Correct Answer: 81