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

1 Answer
dhruv ahlawat
Mar 8, 2018

the antiderivative of #f(x) = e^x - e^-(2x) #

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

that is to say that the derivative of #F(x)# is equal to #f(x)#
#= d/dx (e^x - e^(2x) / 2 + C) = e^x - e^(-2x) #

now we just need to find C to get the total antiderivative of #f(x)#
and we can use the fact that at #x = 0# the value comes out as -2
#therefore f( 0) = -2#

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

therefore,
#C = -2 1/2 #
=
therefore, the entire function becomes
#e^x - e^(2x) / 2 - 2 1/2#

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