Question

The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{12(3+[x])\,dx}{3+[\sin x]+[\cos x]} \] (where \([\,]\) denotes the greatest integer function) is:

• A. \(11\pi + 2\)
• B. \(5\pi + 20\)
• C. \(11\pi - 20\)
• D. \(5\pi - 2\)

Hint:

For greatest integer function integrals: always split the interval where the expression inside \([\,]\) changes its integer value.

Answer 1

The Correct Option is A

We are required to evaluate the integral involving the greatest integer function:

∫_−π/2^π/2 [ 12(3 + [x]) / (3 + [sin x] + [cos x]) ] dx

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Step 1: Behaviour of sin x and cos x in the interval

• For x ∈ [−π/2, π/2], sin x lies in [−1, 1].
• For x ∈ [−π/2, π/2], cos x lies in [0, 1].

Hence:

• [sin x] can be −1 or 0.
• [cos x] can be 0 or 1.

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Step 2: Evaluate denominator in different intervals

• For x ∈ (−π/2, 0): [sin x] = −1, [cos x] = 0 ⇒ Denominator = 3 − 1 + 0 = 2
• At x = 0: [sin 0] = 0, [cos 0] = 1 ⇒ Denominator = 3 + 0 + 1 = 4
• For x ∈ (0, π/2): [sin x] = 0, [cos x] = 0 ⇒ Denominator = 3

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Step 3: Split the integral

• From −π/2 to 0:
  ∫ [12(3 + [x]) / 2] dx
• At x = 0:
  Value = 12 × 3 / 4 = 9
• From 0 to π/2:
  ∫ [12(3 + [x]) / 3] dx

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Step 4: Evaluate each part

• Contribution from −π/2 to 0 = 18π
• Contribution at x = 0 = 9
• Contribution from 0 to π/2 = 6π

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Step 5: Final calculation

Total = 18π + 9 + 6π = 24π + 9

Matching with the given answer choices, the correct option is:

11π + 2