If = ₀^ 2 3 ₂(x^2 + 4) ,dx + ₂^4 2^x - 4 ,dx , then ^2 is equal to _______.

Step-by-step solution

Let f(x) = ₂(x^2 + 4) for x 0 . To find the inverse function f⁻¹(x) , we set y = ₂(x^2 + 4) and solve for x : 2^y = x^2 + 4 x^2 = 2^y - 4 x = 2^y - 4 Thus, f⁻¹(x) = 2^x - 4 . The given expression is = ₀^ 2 3 f(x) ,dx + ₂^4 f⁻¹(x) ,dx . We observe the limits of the first integral and evaluate the function at these points: f(0) = ₂(0 + 4) = 2 f(2 3 ) = ₂((2 3 )^2 + 4) = ₂(12 + 4) = ₂(16) = 4 These exactly match the limits of the second integral. Using the standard property of definite integrals for inverse functions: