Let f be a polynomial function such that f( x + 1) = x + 2 x + 4 for all x 0 . Then the value of the integral _ -3 ³ f(x) 1 + 2^x dx is equal to

Step-by-step solution

Let u = x + 1 , which implies x = u - 1 and x = (u - 1)^2 . Substitute this into the given functional equation: f(u) = (u - 1)^2 + 2(u - 1) + 4 f(u) = u^2 - 2u + 1 + 2u - 2 + 4 f(u) = u^2 + 3 Thus, the function is f(x) = x^2 + 3 . The integral to be evaluated is: I = _ -3 ³ x^2 + 3 1 + 2^x dx Using the definite integral property _ -a ^ a g(x) 1 + b^x dx = ₀^ a g(x) dx for an even function g(x) , and noting that g(x) = x^2 + 3 is an even function, we can simplify the integral: I = ₀³ (x^2 + 3) dx Evaluating this int