Let f be a polynomial function such that f(x^2 - 2) = x^4 - 4x^2 + 5 for all x R . If ₀^ a f(x) dx = 12 and a > 0 , then the value of a is

Step-by-step solution

Let u = x^2 - 2 , which implies x^2 = u + 2 . Substituting this into the given functional equation: f(u) = (u + 2)^2 - 4(u + 2) + 5 f(u) = u^2 + 4u + 4 - 4u - 8 + 5 f(u) = u^2 + 1 Therefore, the polynomial function is f(x) = x^2 + 1 . Now, evaluate the definite integral: ₀^ a (x^2 + 1) dx = [ x^3 3 + x ]₀^ a = a^3 3 + a We are given that the integral evaluates to 12 : a^3 3 + a = 12 a^3 + 3a - 36 = 0 By inspection, for a = 3 , we get 3^3 + 3(3) - 36 = 27 + 9 - 36 = 0 . Since a^3 + 3a - 36 = (a - 3)(a^2 + 3a + 12) =