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

1 Answer
Jim H
Jun 8, 2017

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

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