Question

求解定积分：x²e^(-2x)dx,范围：0到正无穷

Answer 1

∫[0,+∞] x²e^(-2x)dx

= -1/2* ∫[0,+∞] x²e^(-2x)d(-2x)

= -1/2* ∫[0,+∞] x² de^(-2x)

= -1/2* { e^(-2x)*x²|[0,+∞] -∫[0,+∞] 2x*e^(-2x)dx }

= -1/2* { 0 +1/2∫[0,+∞] 2x*e^(-2x)d(-2x) }

= -1/2∫[0,+∞] x*de^(-2x)

= -1/2* { e^(-2x)*x|[0,+∞] -∫[0,+∞] e^(-2x)dx }

= -1/2* { 0 +1/2∫[0,+∞] e^(-2x)d(-2x) }

= -1/4*e^(-2x)|[0,+∞]

= 1/4