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
To evaluate the integral x^2 (x)\,dx, we will use integration by parts, which states u\,dv = uv - v\,du. We will need to apply this method twice. Step 1: First application of integration by parts. Choose u and dv: Let u = x^2 Then du = 2x\,dx Let dv = (x)\,dx Then v = (x)\,dx = (x) Apply the integration by parts formula: x^2 (x)\,dx = x^2 (x) - (x) (2x)\,dx x^2 (x)\,dx = x^2 (x) - 2 x (x)\,dx Step 2: Second application of integration by parts for the integral x (x)\,dx. Choose new u and dv: Let u_1 = x Then du_1 = dx Let dv_1 = (x)\,dx Then v_1 = (x)\,dx = -(x) Apply the integration by parts formula to x (x)\,dx: x (x)\,dx = x(-(x)) - (-(x))\,dx x (x)\,dx = -x(x) + (x)\,dx x (x)\,dx = -x(x) + (x) Step 3: Substitute the result from Step 2 back into the equation from Step 1. x^2 (x)\,dx = x^2 (x) - 2 (-x(x) + (x)) + C x^2 (x)\,dx = x^2 (x) + 2x(x) - 2(x) + C The final answer is x^2 (x) + 2x(x) - 2(x) + C. That's 2 down. 3 left today — send the next one.