What is #F(x) = int 3x^2+e^(2-2x) dx# if #F(0) = 1 #?

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Answer 1 · 2017-01-12T08:46:29

#F(x) = x^3 +e^2/2 (1-e^(-2x)) +1#

#int 3x^2+e^(2-2x) dx = 3intx^2 dx+ e^2 int e^(-2x) dx#

#= 3*x^3/3 + e^2 int e^(-2x) dx#

#int e^(-2x) dx = -1/2e^(-2x)#

#:. F(x) = x^3 - e^2/2 * e^(-2x) +C #

Since #F(0) =1#

# 0^3 - e^2/2 * e^(-2*0) +C =1#

#-e^2/2 *1 +C =1#

#C= 1+e^2/2#

#F(x) = x^3 - e^2/2 * e^(-2x) +1+e^2/2#

#= x^3 +e^2/2 (1-e^(-2x)) +1#