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

1 Answer
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