What is #F(x) = int e^(x-2) - x dx# if #F(0) = 1 #?

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Answer 1 · 2017-04-28T10:27:35

# F(x) = e^(x-2) - 1/2x^2 + 1 - e^(-2) #

We have:

# F(x) = int \ e^(x-2) - x \ dx #

Integrating wrt #x# we get:

# F(x) = e^(x-2) - 1/2x^2 + C #

But we know that #F(0)=1#, and so:

# e^(0-2) - 0 + C = 1#
# :. e^(-2) + C = 1#
# :. C = 1 - e^(-2)#

Hence,

# F(x) = e^(x-2) - 1/2x^2 + 1 - e^(-2) #