What is #F(x) = int xe^(-x) dx# if #F(0) = 2 #?

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Answer 1 · 2016-09-13T20:46:50

# F(x) = 3 - e^(-x) (x+1) #

#F(x) = int xe^(-x) dx#

# = int xd/dx(-e^(-x)) dx#

# = x(-e^(-x)) + int d/dx(x) e^(-x) dx + C#

# = -x e^(-x) + int e^(-x) dx + C#

# = -x e^(-x) - e^(-x) +C#

#\implies F(x) = - e^(-x) (x+1) +C#

#F(0) = 2 implies 2 = - 1 + C#

#\implies F(x) = 3 - e^(-x) (x+1) #