y＾2×4e＾（-4y）在（0,∞）上对y积分

Question

y＾2×4e＾（-4y）在（0, ∞）上对y积分

Answer 1

先求不定积分
∫y^2*4e^(-4y)dy = -∫y^2de^(-4y) = -y^2*e^(-4y)+∫e^(-4y)2ydy
= -y^2*e^(-4y)-(1/2)∫yde^(-4y)
= -y^2*e^(-4y)-(1/2)ye^(-4y)+(1/2)∫e^(-4y)dy
= -y^2*e^(-4y)-(1/2)ye^(-4y)-(1/8)e^(-4y)+C
=-(y^2+y/2+1/8)e^(-4y)+C.
lim[-(y^2+y/2+1/8)e^(-4y)]
= lim[-(y^2+y/2+1/8)/e^(4y)]=0 （用2次罗必塔法则得出）
则 ∫y^2*4e^(-4y)dy =[-(y^2+y/2+1/8)e^(-4y)] = 1/8.