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Consider the definite integrals A = ∫ 0 π x sin ⁡ x cos 2 ⁡ x d x and B = ∫ 0 π 2 sin ⁡ x cos 2 ⁡ x d x . Then,

Options

  1. AA = 2 B
  2. BA = π B
  3. CA = π 2 B
  4. DB = 2 A

Correct answer

B. A = π B

Step-by-step solution

Applying a + b - x property in A , and adding, we get, 2 A = ∫ 0 π x + π - x sin ⁡ x cos 2 ⁡ x d x ⇒ 2 A = π ∫ 0 π sin ⁡ x cos 2 ⁡ x d x ∴ 2 A = π ⋅ 2 ∫ 0 π 2 sin x cos 2 x d x ⇒ A = π B

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