Question

We have `F:RR`*`->RR` such that `F'(x)=1/x,F(-1)=1,F(1)=0`.How to calculate `F(e)+F(-e)`?

Answer 1

The antiderivative has two parts (pieces, cases).

Explanation:

From the fact that we are given `F(-1)` and `F(1)`, I conclude that the domain includes positive and negative real numbers.

`F'` however has a discontinuity at `0`, so we must allow that the antiderivative may be defined by cases (piecewise) with different constants on each piece.

From `F'(x) = 1/x`, we conclude that

`F(x) = {(lnx+C_1,x > 0),(ln(-x)+C_2,x < 0):}`

Now use `F(-1) = 1` to conclude that `C_2 = 1`

and `F(1) = 0` tells us that `C_1 = 0`.

So we have

`F(x) = {(lnx, x > 0),(ln(-x)+1,x < 0):}`

Finally,

`F(e)+F(-e) = ln(e)+(ln(e)+1) = 3`