Question

What is `f(x) = int e^(3x)-e^(x)dx` if `f(0)=-2`?

Answer 1

I found: `f(x)=e^(3x)/3-e^x-4/3`

Explanation:

We can solve the integral and write:
`f(x)=inte^(3x)dx-inte^xdx=e^(3x)/3-e^x+c`
now we need to find `c`;
we use the fact that `f(0)=-2`, i.e., we use `x=0` and set it equal to `-2`:
`e^(3*0)/3-e^0+c=-2`
`1/3-1+c=-2`
`c=1-2-1/3=-4/3`
giving your final function as:
`f(x)=e^(3x)/3-e^x-4/3`