Definite Integration
Properties of definite integrals
Grade 12
Question:
<p>Let \(I_n = \int_{-\pi}^{\pi} \frac{1}{1+2^{\sin\left(\frac{x}{2}\right)}} \left(\frac{\sin\left(\frac{nx}{2}\right)}{\sin\left(\frac{x}{2}\right)}\right)^2 dx\), for \(n = 0, 1, 2, 3, \ldots\), then which of the following is/are always correct?</p>
<p>(a) \(I_{n+1} - I_n = \pi \; \forall n = 0, 1, 2, 3, \ldots\)</p>
<p>(b) \(I_0, I_1, I_2, I_3, \ldots, I_n\) form an A.P.</p>
<p>(c) \(\displaystyle\sum_{m=0}^{9} I_{2m} = 90\pi\)</p>
<p>(d) \(\displaystyle\sum_{m=0}^{10} I_m = 65\pi\)</p>
Step-by-Step Solution
Key Concept: Use the property that f(x) + f(-x) can be evaluated by substituting x → -x, and recognize that the integrand structure allows decomposition using symmetry properties of even/odd functions combined with the Dirichlet kernel properties.
<p><strong>Step 1: Apply substitution x → -x</strong></p><p>Let J = ∫_{-π}^{π} [1/(1+2^(sin(x/2))) · (sin(nx/2)/sin(x/2))^2] dx</p><p>Substitute x → -x: J = ∫_{-π}^{π} [1/(1+2^(-sin(x/2))) · (sin(-nx/2)/sin(-x/2))^2] dx</p><p>Since (sin(-u)/sin(-u))^2 = (sin(u)/sin(u))^2, the Dirichlet kernel part is unchanged.</p><p><strong>Step 2: Simplify the denominator</strong></p><p>Note that: 1/(1+2^(-sin(x/2))) = 2^(sin(x/2))/(1+2^(sin(x/2)))</p><p>Therefore: 1/(1+2^(sin(x/2))) + 2^(sin(x/2))/(1+2^(sin(x/2))) = 1</p><p><strong>Step 3: Add the original and transformed integrals</strong></p><p>I_n + J = ∫_{-π}^{π} [1/(1+2^(sin(x/2))) + 2^(sin(x/2))/(1+2^(sin(x/2)))] · (sin(nx/2)/sin(x/2))^2 dx</p><p>I_n + J = ∫_{-π}^{π} (sin(nx/2)/sin(x/2))^2 dx</p><p>But J = I_n (by the substitution symmetry), so:</p><p>2I_n = ∫_{-π}^{π} (sin(nx/2)/sin(x/2))^2 dx = 2π for all n ≥ 0</p><p><strong>Step 4: Conclusion</strong></p><p>Therefore: I_n = π for all n = 0, 1, 2, 3, ...</p><p>∴ Answer: I_n = π (constant for all n); all statements asserting I_n = π are correct</p>
Correct Answer: ABCD