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JEE Main202127 Jul 2021Morning ShiftMathematicsDefinite IntegrationActual

The value of the definite integral ∫ - π 4 π 4 d x 1 + e x cos x sin 4 x + cos 4 x is equal to :

Options

  1. A- π 2
  2. Bπ 2 2
  3. C- π 4
  4. Dπ 2

Correct answer

B. π 2 2

Step-by-step solution

I = ∫ - π 4 π 4 d x 1 + e x cos x sin 4 x + cos 4 x         … 1 Using ∫ a b f x d x = ∫ a b f a + b - x d x I = ∫ - π / 4 π / 4 d x 1 + e - x cos x sin 4 x + cos 4 x         . . . 2 Add 1 and 2 2 I = ∫ - π 4 π 4 d x sin 4 x + cos 4 x ⇒ 2 I = 2 ∫ 0 π 4 d x sin 4 x + cos 4 x ⇒ I = ∫ 0 π 4 1 + tan 2 x sec 2 x tan 4 x + 1 d x ⇒ I = ∫ 0 π 4 1 + 1 tan 2 x sec 2 x tan

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