Question

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

Answer 1

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`