If _ - 4 ^ 4 ( x^2 2x 1+e^ x + a ^2 x 1+e^ x ) dx = ^2 32 - 1 4 + 3 , then the value of a is

Step-by-step solution

Let I = _ - 4 ^ 4 ( x^2 2x 1+e^ x + a ^2 x 1+e^ x ) dx . Using the property _ -A ^ A f(x) dx = ₀^ A (f(x) + f(-x)) dx , we can simplify the integral. For the first term, let f₁(x) = x^2 2x 1+e^ x . Then f₁(x) + f₁(-x) = x^2 2x 1+e^ x + x^2 2x 1+e^ - x = x^2 2x ( 1 1+e^ x + e^ x e^ x +1 ) = x^2 2x . For the second term, let f₂(x) = a ^2 x 1+e^ x . Then f₂(x) + f₂(-x) = a ^2 x 1+e^ x + a ^2 x 1+e^ - x = a ^2 x . Thus, the integral becomes: I = ₀^ 4 (x^2 2x + a ^2 x) dx Evaluating the first part using integration by p