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: Rewrite the integral and apply integration by parts. Let I = ^3 x \, dx. We can write ^3 x = x · ^2 x. Use integration by parts, u \, dv = uv - v \, du. Let u = x and dv = ^2 x \, dx. Then du = x x \, dx and v = x. I = x x - x ( x x) \, dx I = x x - x ^2 x \, dx Step 2: Use the trigonometric identity ^2 x = ^2 x - 1. Substitute this identity into the integral: I = x x - x (^2 x - 1) \, dx I = x x - (^3 x - x) \, dx I = x x - ^3 x \, dx + x \, dx Step 3: Solve for I. Notice that the original integral I = ^3 x \, dx appears on the right side. I = x x - I + x \, dx Add I to both sides: 2I = x x + x \, dx Step 4: Evaluate the integral of x. The integral of x is | x + x|. 2I = x x + | x + x| + C' Divide by 2 to solve for I: I = (1)/(2) ( x x + | x + x|) + C where C = (C')/(2) is the constant of integration. The final answer is (1)/(2) ( x x + | x + x|) + C.