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JEE Main20266 April 2026Morning ShiftMathematicsDefinite IntegrationActual

The value of the integral _ - /4 ^ /4 ( 32 ^4 x 1 + e^ x )dx is:

Options

  1. A4 + 2
  2. B3 + 8
  3. C3 + 4
  4. D4 + 3

Correct answer

B. 3 + 8

Step-by-step solution

Let I = _ - /4 ^ /4 32 ^4 x 1 + e^ x dx Using the definite integral property _ -a ^ a f(x) dx = ₀^ a (f(x) + f(-x)) dx , we get: I = ₀^ /4 ( 32 ^4 x 1 + e^ x + 32 ^4(-x) 1 + e^ (-x) ) dx I = ₀^ /4 ( 32 ^4 x 1 + e^ x + 32 ^4 x 1 + e^ - x ) dx I = ₀^ /4 ( 32 ^4 x 1 + e^ x + 32 ^4 x e^ x e^ x + 1 ) dx I = ₀^ /4 32 ^4 x ( 1 + e^ x 1 + e^ x ) dx I = ₀^ /4 32 ^4 x dx Using the trigonometric identity ^2 x = 1 + 2x 2 , we can write: ^4 x = ( 1 + 2x 2 )^2 = 1 4 (1 + 2 2x + ^2 2x) ^4 x = 1 4 ( 1 + 2 2x + 1 + 4x 2 ) = 3 8 + 1

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