If the value of the integral _ -1 ^1 e^ x + e^ - x 1+5^x dx is e^2 - e⁻² 2 , then the positive value of is

Step-by-step solution

Let I = _ -1 ^1 e^ x + e^ - x 1+5^x dx Using the property _ -a ^a f(x) dx = ₀^a (f(x) + f(-x)) dx , we get: I = ₀^1 ( e^ x + e^ - x 1+5^x + e^ - x + e^ x 1+5^ -x ) dx I = ₀^1 (e^ x + e^ - x ) ( 1 1+5^x + 5^x 5^x+1 ) dx I = ₀^1 (e^ x + e^ - x ) dx I = [ e^ x - e^ - x ]₀^1 I = e^ - e^ - Given that I = e^2 - e⁻² 2 , we have: e^ - e^ - = e^2 - e⁻² 2 By comparing both sides, we can see that = 2 satisfies the equation. Since the function f( ) = e^ - e^ - is strictly increasing for > 0 , = 2 is the unique positive solutio