Definite Integration
Integration by parts with recurrence relations
Grade 12
Question:
<p>If <i>A</i>(<i>n</i>) represents the area bounded by the curve <i>y</i> = <i>n</i> ln <i>x</i>, where <i>n</i> ∈ ℕ and <i>n</i> > 1, the x-axis and the lines <i>x</i> = 1 and <i>x</i> = e, then the value of <i>A</i>(<i>n</i>) + <i>n</i><i>A</i>(<i>n</i>–1) is equal to:</p>
<p>(A) n</p>
<p>(B) 2</p>
<p>(C) n²</p>
<p>(D) e·n²</p>
Step-by-Step Solution
Key Concept: Compute A(n) = ∫₁ᵉ n ln x dx using integration by parts, establish a recurrence relation, and verify A(n) + nA(n–1) = n.
<p>Not provided in source text</p>
Correct Answer: A