This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the integral and its limits. The integral to evaluate is: _()/(2)^(3)/(2) ^3 x x \, dx Step 2: Use a substitution method. Let u = x. Then, the differential du is du = x \, dx. Step 3: Change the limits of integration based on the substitution. When x = ()/(2): u = (()/(2)) = 1. When x = (3)/(2): u = ((3)/(2)) = -1. Step 4: Rewrite the integral in terms of u and evaluate. Substitute u and du into the integral with the new limits: _1^-1 u^3 \, du Now, integrate u^3 with respect to u: [ u^3+13+1 ]_1^-1 = [ (u^4)/(4) ]_1^-1 Step 5: Apply the limits of integration. ((-1)^4)/(4) - ((1)^4)/(4) = (1)/(4) - (1)/(4) = 0 The value of the definite integral is 0. 0 3 done, 2 left today. You're making progress.