Question

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

Answer 1

`f(x)=(3-x)e^x+e^x-6`

Explanation:

Given `f(x)=int(3-x)e^xdx` and that `f(0)=-2`.

Now, in the given function, we can solve using the formula
`int(uv)dx=uintvdx-int(frac{d}{dx}(u)*intvdx)dx`
Takin `u=3-x` and `v=e^x` you'll end up with
`int(3-x)e^xdx=(3-x)inte^xdx-int(d/dx(3-x)*inte^xdx)dx`
So the answer if you solve the above is `f(x)=(3-x)e^x+e^x+c`

Now, they have also said that `f(0)=-2` which means we're going to have to find out what that `c` is.
So `f(0)=(3-0)e^0+e^0+cimplies-2=3+1+cimpliesc=-6`

So, taking it as such, we get the answer as given above.