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

1 Answer
Sep 13, 2016

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

Explanation:

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)