Definite Integration
Evaluation of Definite Integrals
Grade 12
Question:
<p>If \(I_n = \displaystyle\int_{-1}^{1} |x|\left(1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + \ldots + \dfrac{x^{2n}}{2n}\right)dx\), then:</p>
<p>(a) \(I_2 = \dfrac{4}{3}\)</p>
<p>(b) \(I_2 = \dfrac{7}{6}\)</p>
<p>(c) \(\displaystyle\lim_{n \to \infty} I_n = \dfrac{3}{2}\)</p>
<p>(d) \(\displaystyle\lim_{n \to \infty} I_n = \dfrac{5}{4}\)</p>
Step-by-Step Solution
Key Concept: Split the integral at x=0 using |x|=-x for x<0 and |x|=x for x>0, then use symmetry properties. The even part of the integrand contributes while odd parts vanish.
<p><strong>Step 1:</strong> Split integral using properties of |x|:</p><p>$I_n = \int_{-1}^{0} (-x)\left(1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots + \frac{x^{2n}}{2n}\right)dx + \int_{0}^{1} x\left(1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots + \frac{x^{2n}}{2n}\right)dx$</p><p><strong>Step 2:</strong> Identify even and odd parts. The integrand $x · P(x)$ where $P(x)$ contains even powers (contribute) and odd powers (vanish over symmetric interval):</p><p>$I_n = 2\int_{0}^{1} x\left(1 + \frac{x^2}{2} + \frac{x^4}{4} + \ldots + \frac{x^{2n}}{2n}\right)dx$</p><p><strong>Step 3:</strong> Integrate term-by-term:</p><p>$I_n = 2\int_{0}^{1} \left(x + \frac{x^3}{2} + \frac{x^5}{4} + \ldots + \frac{x^{2n+1}}{2n}\right)dx$</p><p>$= 2\left[\frac{x^2}{2} + \frac{x^4}{8} + \frac{x^6}{24} + \ldots + \frac{x^{2n+2}}{2n(2n+2)}\right]_0^1$</p><p>$= 2\left(\frac{1}{2} + \frac{1}{8} + \frac{1}{24} + \ldots + \frac{1}{2n(2n+2)}\right)$</p><p>$= 1 + \frac{1}{4} + \frac{1}{12} + \ldots + \frac{1}{n(2n+1)}$</p><p>∴ Answer: B</p>
Correct Answer: B