再一次求不定积分,万分感激.∫x^2e^(-2x)dx详细步骤,非常感激.

Question

再一次求不定积分,万分感激.
∫x^2e^(-2x)dx详细步骤,非常感激.

Answer 1

∫ x²e^(-2x) dx
= (-1/2)∫ x² de^(-2x)
= (-1/2)x²e^(-2x) + (1/2)∫ e^(-2x) dx²,分部积分
= (-1/2)x²e^(-2x) + ∫ xe^(-2x) dx
= (-1/2)x²e^(-2x) - (1/2)∫ x de^(-2x)
= (-1/2)x²e^(-2x) - (1/2)xe^(-2x) + (1/2)∫ e^(-2x) dx,分部积分
= (-1/2)x²e^(-2x) - (1/2)xe^(-2x) + (1/2)(-1/2)e^(-2x) + C
= (-1/2)x²e^(-2x) - (1/2)xe^(-2x) - (1/4)e^(-2x) + C
= (-1/4)(2x² + 2x + 1)e^(-2x) + C