In this post I use the derivatives of f(x) to create a new function, F(x), that will help us evaluate the integral from part 1.

Table of Contents

1. Outline

2. A polynomial and its derivatives

3. A new function

4. The integral is an integer

5. The size of the integral (pending)

We define the new function as the alternating sum of the even derivatives of *f(x)*. In symbols,

Using sigma notation, we can write this more compactly as

First, note that *F(0)* and *F(π)* are both integers. We have already determined that all derivatives of *f(x)* at 0 and π are integers, so any sum or difference of those derivatives must also be an integer. *F(0)* and *F(π)* are such sums or differences; thus, they must be integers.

Now consider *F”(x)+F(x)*. Note from the definition of *F* that

The final term should techinically become

but as we have noted previously, this is zero, so we ignore it. Note that every term in *F(x)* has the opposite sign of the corresponding term of *F”(x)*, with the exception of *f(x)*, which does not have a corresponding term in *F”(x)*. Thus,

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