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JEE MainMathematicsDefinite Integration

The value of the integral _ - /4 ^ /4 ^2 x (2 - 2x)(1 + e^x) dx is :

Options

  1. A1 + 2 3 3
  2. B1 3 + 2 9 ⁻¹(3)
  3. C2 3 + 4 9 3
  4. D1 3 + 2 9 3

Correct answer

D. 1 3 + 2 9 3

Step-by-step solution

Let I = _ - /4 ^ /4 ^2 x (2 - 2x)(1 + e^x) dx Using the property _ -a ^ a f(x) dx = ₀^ a (f(x) + f(-x)) dx , we have: f(x) + f(-x) = ^2 x 2 - 2x ( 1 1 + e^x + 1 1 + e^ -x ) Since 1 1 + e^x + 1 1 + e^ -x = 1 1 + e^x + e^x e^x + 1 = 1 , the expression simplifies to: f(x) + f(-x) = ^2 x 2 - 2x Thus, I = ₀^ /4 ^2 x 2 - 2x dx Substitute 2x = 1 - ^2 x 1 + ^2 x : 2 - 2x = 2 - 1 - ^2 x 1 + ^2 x = 2 + 2 ^2 x - 1 + ^2 x 1 + ^2 x = 1 + 3 ^2 x 1 + ^2 x Substituting this back into the integral: I = ₀^ /4 ^2 x (1 + ^2 x) 1 + 3 ^

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