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.
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You're on a roll — to evaluate the integral x^2 e^-x\,dx, we will use integration by parts, which states u\,dv = uv - v\,du. We will 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 = e^-x\,dx Then v = e^-x\,dx = -e^-x Apply the integration by parts formula: x^2 e^-x\,dx = x^2(-e^-x) - (-e^-x)(2x)\,dx x^2 e^-x\,dx = -x^2 e^-x + 2 x e^-x\,dx Step 2: Second application of integration by parts for the integral x e^-x\,dx. Choose new u and dv: Let u_1 = x Then du_1 = dx Let dv_1 = e^-x\,dx Then v_1 = e^-x\,dx = -e^-x Apply the integration by parts formula to x e^-x\,dx: x e^-x\,dx = x(-e^-x) - (-e^-x)\,dx x e^-x\,dx = -x e^-x + e^-x\,dx x e^-x\,dx = -x e^-x - e^-x Step 3: Substitute the result from Step 2 back into the equation from Step 1. x^2 e^-x\,dx = -x^2 e^-x + 2 (-x e^-x - e^-x) + C x^2 e^-x\,dx = -x^2 e^-x - 2x e^-x - 2 e^-x + C We can factor out -e^-x: x^2 e^-x\,dx = -e^-x(x^2 + 2x + 2) + C The final answer is -e^-x(x^2 + 2x + 2) + C. One more to go today! Send it over.