Final answer:
The piecewise integral must be evaluated in two parts due to its definition. The expression for the separate integral with the function x^(-3) is incorrect because the function is not continuous on the interval [-3, 1], which precludes the application of the second part of the Fundamental Theorem of Calculus.
Step-by-step explanation:
You are given two pieces of information in your math problem: a definite integral of a piecewise function and a separate integral calculation that you have concerns about. First, let's address the piecewise function integral. To evaluate the integral π∫ f(x) dx where f(x) is defined as 2 sin x for 0 < x < π/2 and 3 cos x for π/2 ≤ x ≤ π, you need to split the integral into two at x = π/2 and evaluate each integral separately, then sum the results.
For the second integral ∫ x⁻³ dx from -3 to 1, there is an issue. The correct answer is (d): f(x) = x⁻³ is not continuous on the interval [-3, 1] so FTC2 (the second part of the Fundamental Theorem of Calculus) cannot be applied. Specifically, the function x⁻³ is not defined at x = 0, which makes the function discontinuous on the interval, hence the problem with the integral equation given. The function must be continuous on the entire interval for FTC2 to be applicable.