求定积分:(x-1)^2*e^(-x)积分限x从0到正无穷

Question

求定积分:(x-1)^2 * e^(-x) 积分限x从0到正无穷

Answer 1

结果是1.
∫ (x-1)² * e^-x dx
= -∫ (x-1)² de^-x
= -(x-1)² * e^-x + ∫ e^-x * 2(x-1) dx
= -(x²-2x+1) * e^-x - 2∫ (x-1) de^-x
= (-x²e^-x + 2xe^-x - e^-x) - 2(x-1) * e^-x + 2∫ e^-x dx
= -x²e^-x + 2xe^-x - e^-x - 2xe^-x + 2e^-x - 2e^-x + C
= -x²e^-x - e^-x + C
= -(x²+1) / e^x + C
∴∫(0->+∞) (x-1)² * e^-x dx
= lim[x->+∞] -(x²+1) / e^x - lim[x->0] -(x²+1) / e^x
= -2lim[x->+∞] x / e^x + (0+1) / e^0
= -2lim[x->∞] 1 / e^x + 1
= -2 * 0 + 1
= 1