<p><strong>311.</strong> Let \(\displaystyle\int \dfrac{(x-1)e^x}{(x+1)^3}\, dx = f(x) + C\) where \(f(x) = d + \displaystyle\sum_{i=0}^{n} \dfrac{a_i e^x}{(x+1)^i}\) with all \(a_i = 0\) for \(i \geq n\) and \(f(1) = \dfrac{e}{2}\). Then which of the following is/are correct?</p>
Step-by-Step Solution
Key Concept: Decompose the integrand using partial fractions after rewriting the numerator in terms of (x+1), then integrate term-by-term. The condition f(1) = e/2 determines the constant d.
<p><strong>Step 1:</strong> Rewrite the numerator: x - 1 = (x+1) - 2</p><p>∴ ∫[(x-1)e^x/(x+1)³]dx = ∫[((x+1)-2)e^x/(x+1)³]dx = ∫[e^x/(x+1)²]dx - 2∫[e^x/(x+1)³]dx</p><p><strong>Step 2:</strong> Use integration by parts on each integral.</p><p>For ∫[e^x/(x+1)²]dx: Let u = 1/(x+1)², dv = e^x dx</p><p>= e^x/(x+1)² + 2∫[e^x/(x+1)³]dx</p><p><strong>Step 3:</strong> Substitute back:</p><p>∫[(x-1)e^x/(x+1)³]dx = e^x/(x+1)² + 2∫[e^x/(x+1)³]dx - 2∫[e^x/(x+1)³]dx = e^x/(x+1)² + C</p><p><strong>Step 4:</strong> Thus f(x) = e^x/(x+1)² + d</p><p><strong>Step 5:</strong> Apply f(1) = e/2: e¹/(1+1)² + d = e/2</p><p>e/4 + d = e/2</p><p>∴ d = e/4</p><p><strong>Step 6:</strong> Therefore f(x) = e^x/(x+1)² + e/4, which matches the form with a₂ = 1, a₀ = 0, a₁ = 0, a₃ = 0, n = 2, and d = e/4</p>
Correct Answer: ABC