Question

What is `f(x) = int 3x^3+xe^(x-2)+e^x dx` if `f(2 ) = 4`?

Answer 1

I found: `f(x)=3/4x^4+e^(x-2)(x-1)+e^x-9-e^2`

Explanation:

Let us first solve our integral using also integration by parts on `xe^(x-2)`:
`f(x)=int3x^3dx+intxe^(x-2)dx+inte^xdx`
`f(x)=3x^4/4+xe^(x-2)-inte^(x-2)dx+e^x`
`f(x)=3/4x^4+xe^(x-2)-e^(x-2)+e^x+c`
`f(x)=3/4x^4+e^(x-2)(x-1)+e^x+c`
now we evaluate it at `x=2` to get:
`f(2)=3/4(2^4)+e^(2-2)(2-1)+e^2+c`
`f(2)=12+1+e^2+c`
`f(2)=13+e^2+c`

we use the fact that `f(2)=4` to find `c` and write:
`13+e^2+c=4`
so that rearranging:
`c=-9-e^2`

and our function will be:

`f(x)=3/4x^4+e^(x-2)(x-1)+e^x-9-e^2`