Question

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

Answer 1

The answer consists in calculating the primitive `f` (antiderivative) of the given function, and then evaluating the constant for which the primitive `f(2) = 3`

Explanation:

The integral distributes with respect to the sum, so:
`int (e^(x+2) + x) dx = int e^(x+2) dx + int x dx`

So, solving the first one:
`int e^(x+2) dx = int e^x e^2 dx = e^2 int e^x dx= e^2 e^x + C_1` where `C_1` is an unknown constant.

Similarly, the second one is:
`int x dx = x^2/2 + C_2`

So the antiderivative of `e^(x+2) + x` is:
`f(x) = e^2 e^x + x^2/2 + C`, where C is a constant.

Now we have to solve for `f(2) = 3`:
`e^2 e^2 + 2^2/2 + C = 3`, that is:
`e^4 + 2 + C = 3`, thus:
`e^4 + C = 1`, and then
`C = 1-e^4`.

The final answer is then:
`f(x) = e^2 e^x + x^2/2 + (1-e^4)`