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JEE Main202224 Jun 2022Evening ShiftMathematicsDefinite IntegrationActual

The value of the integral ∫ - π 2 π 2 d x 1 + e x sin 6 x + cos 6 x is equal to

Options

  1. A2 π
  2. B0
  3. Cπ
  4. Dπ 2

Correct answer

C. π

Step-by-step solution

Let, I = ∫ - π 2 π 2 d x 1 + e x sm 6 x + cos 6 x                 . . . i Now we know ∫ a b f x d x = ∫ a b f a + b - x d x I = ∫ - π 2 π 2 e x d x 1 + e x sin 6 x + cos 6 x         . . . ii Adding i and ii 2 I = ∫ - π 2 π 2 1 + e x d x 1 + e x sin 6 x + cos 6 x I = 1 2 × 2 ∫ 0 π 2 d x sin 2 x + cos 2 x sin 4 x + cos 4 x - sin 2 x cos 2 x = ∫ 0 π 2 d x sin 2 x + cos 2 x

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