∫xe^(-x)dx=?从1积到0∫xe^(-x)dx=?从2积到正无穷

Question

∫xe^(-x) dx=?从1积到0
∫xe^(-x) dx=?从2积到正无穷

Answer 1

∫xe^(-x) dx
=-xe^(-x) -∫(-x)'e^(-x) dx
=-xe^(-x)+∫e^(-x) dx
=-xe^(-x)-e^(-x)
=-(x+1)e^(-x)
从1积到0
[-(0+1)e^0]-[-(1+1)e^(-1)]
=-1+2/e
=(2/e)-1
从2积分到正无穷
[-(正无穷+1)*e^-无穷]-[-(2+1)e^(-2)]
=0+3/e²
=3/e²