Definite Integration Question - None

Question

The value of $\displaystyle\int_{-\pi/2}^{\pi/2} \frac{dx}{[x]+[\sin x]+4}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is:

Accepted Answer

Step 1: Analyze the integrand f(x) = 1/([x] + [sin x] + 4)

For x ∈ [-π/2, π/2], we have -1 ≤ sin x ≤ 1, so [sin x] ∈ {-1, 0}.

Step 2: Check if f(x) is even. For x ∈ (0, π/2):

• [x] = 0 and [sin x] = 0, so f(x) = 1/4

For x ∈ (-π/2, 0):

• [x] = -1 and [sin x] = -1, so f(x) = 1/(-1 - 1 + 4) = 1/2

Step 3: Note f(-x) ≠ f(x), but split the integral using symmetry properties:

∫_{-π/2}^{π/2} = ∫_{-π/2}^{0} + ∫_{0}^{π/2}

Step 4: For x ∈ (0, π/2): [x] = 0, [sin x] = 0 ⟹ integrand = 1/4

∫_{0}^{π/2} dx/4 = π/8

Step 5: For x ∈ (-π/2, 0): [x] = -1, [sin x] = -1 ⟹ integrand = 1/2

∫_{-π/2}^{0} dx/2 = π/4

Step 6: Handle boundary points x = 0, ±π/2 (measure zero, don't affect integral)

Total = π/8 + π/4 = 3π/8

∴ Answer: C

Question: