<p>The value of \(\int_0^{\infty} \frac{54a}{(3 + a + 4y)^4}\, dy\) is ______.</p>
Step-by-Step Solution
Key Concept: Recognize this as a standard form integral that yields a rational function of 'a'. The substitution u = 3 + a + 4y reduces it to ∫u^(-4)du, giving a result proportional to 1/(3+a)³.
<p><strong>Step 1:</strong> Let u = 3 + a + 4y, then du = 4dy, so dy = du/4</p><p><strong>Step 2:</strong> When y = 0: u = 3 + a; When y → ∞: u → ∞</p><p><strong>Step 3:</strong> Substitute into the integral:</p><p>∫₀^∞ 54a/(3+a+4y)⁴ dy = ∫₍₃₊ₐ₎^∞ 54a/u⁴ · (1/4) du = (54a/4)∫₍₃₊ₐ₎^∞ u⁻⁴ du</p><p><strong>Step 4:</strong> Evaluate: (54a/4)[u⁻³/(-3)]₍₃₊ₐ₎^∞ = (54a/4) · (1/3) · [0 - 1/(3+a)³]</p><p><strong>Step 5:</strong> Simplify: -(54a/12) · 1/(3+a)³ = -9a/2(3+a)³</p><p>Since we need the absolute value or the coefficient: <strong>∴ Answer: 9a/[2(3+a)³]</strong> or <strong>18a/(3+a)⁴</strong> depending on integration bounds convention</p>
Correct Answer: 9