Answer:
17
Step-by-step explanation:
∫ x² e⁻³ˣ dx = -1/27 e⁻³ˣ [Ax² + Bx + E] + C
Take derivative of both sides:
x² e⁻³ˣ = d/dx {-1/27 e⁻³ˣ [Ax² + Bx + E] + C}
x² e⁻³ˣ = -1/27 d/dx {e⁻³ˣ [Ax² + Bx + E]}
-27x² e⁻³ˣ = d/dx {e⁻³ˣ [Ax² + Bx + E]}
Use product rule to evaluate the derivative:
-27x² e⁻³ˣ = {e⁻³ˣ [2Ax + B] − 3e⁻³ˣ [Ax² + Bx + E]}
-27x² e⁻³ˣ = e⁻³ˣ {2Ax + B − 3 [Ax² + Bx + E]}
-27x² e⁻³ˣ = e⁻³ˣ [2Ax + B − 3Ax² − 3Bx − 3E]
-27x² e⁻³ˣ = e⁻³ˣ [-3Ax² + (2A − 3B) x + (B − 3E)]
-27x² = -3Ax² + (2A − 3B) x + (B − 3E)
Match the coefficients:
-27 = -3A
0 = 2A − 3B
0 = B − 3E
Solve the system of equations:
A = 9
B = 6
E = 2
Therefore, A + B + E = 17.
