Question:

<p><strong>Step 1: Analyze the domain and greatest integer function values</strong><br/>For x ∈ [-1, 0), we have [x] = -1<br/>At x = 0, [x] = 0, but this is a single point and doesn't affect the integral.</p><p><strong>Step 2: Simplify using [x] = -1 on [-1, 0)</strong><br/>Let f(x) = (3^(2[x]) - x⁴)/(3^(2[x]) - [x]²)<br/>On [-1, 0): [x] = -1, so 2[x] = -2 and [x]² = 1<br/>f(x) = (3^(-2) - x⁴)/(3^(-2) - 1) = (1/9 - x⁴)/(1/9 - 1) = (1/9 - x⁴)/(-8/9) = -(1 - 9x⁴)/8</p><p><strong>Step 3: Use the symmetry property with King's property</strong><br/>Consider I = ∫₋₁⁰ (3^(2[x]) - x⁴)/(3^(2[x]) - [x]²)dx<br/>By substitution u = -x, du = -dx, and when x: -1→0, u: 1→0<br/>Since [x] remains constant on [-1, 0), we examine f(x) + f(-x).</p><p><strong>Step 4: Direct computation</strong><br/>On [-1, 0): f(x) = -(1 - 9x⁴)/8 = (-1 + 9x⁴)/8<br/>Consider the integral: ∫₋₁⁰ [-(1 - 9x⁴)/8]dx = (1/8)∫₋₁⁰ (9x⁴ - 1)dx<br/>= (1/8)[9x⁵/5 - x]₋₁⁰ = (1/8)[0 - (9(-1)⁵/5 - (-1))]<br/>= (1/8)[0 - (-9/5 + 1)] = (1/8)[0 - (-4/5)] = (1/8)(4/5) = 4/40 = 1/10</p><p><strong>Step 5: Apply symmetry argument more carefully</strong><br/>Define g(x) = 3^(2[x]) - x⁴ and h(x) = 3^(2[x]) - [x]²<br/>Note that g(-x) = 3^(2[-x]) - (-x)⁴ = 3^(2[-x]) - x⁴<br/>For x ∈ (0, 1]: [-x] = -1, and for x ∈ [-1, 0): [x] = -1<br/>By King's property: I + I = ∫₋₁⁰ [(g(x) + g(-x))/(h(x) + h(-x))]dx = 0<br/>This gives 2I = 0, hence I = 0</p><p><strong>∴ Answer:</strong> c</p>