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: Use integration by parts. Let u = x and dv = ^2 x \, dx. Then du = x x \, dx and v = x. Step 2: Apply the integration by parts formula: u \, dv = uv - v \, du. ^3 x \, dx = x x - x ( x x) \, dx ^3 x \, dx = x x - x ^2 x \, dx Step 3: Use the identity ^2 x = ^2 x - 1. ^3 x \, dx = x x - x (^2 x - 1) \, dx ^3 x \, dx = x x - (^3 x - x) \, dx Step 4: Distribute the integral. ^3 x \, dx = x x - ^3 x \, dx + x \, dx Step 5: Let I = ^3 x \, dx. The equation becomes: I = x x - I + x \, dx Step 6: Add I to both sides. 2I = x x + x \, dx Step 7: Evaluate the integral of x. x \, dx = | x + x| + C Step 8: Substitute this back into the equation for 2I. 2I = x x + | x + x| + C Step 9: Solve for I. I = (1)/(2) ( x x + | x + x|) + C The final answer is (1)/(2)( x x + | x + x|) + C.