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# Prove formulas for the integral from -b to b of even and odd functions

We define an even function to be a function such that whenever is in the domain of so is , and for all in the domain, we have We define an odd function to be a function such that whenever is in the domain of so is , and for all in the domain, we have Then, for integrable on prove the following.

1. If is an even function then 2. If is an odd function then 1. Proof. First, since the integral is additive with respect to the interval of integration (Theorem 1.17 of Apostol), we have Then, for the first integral we use the expansion/contraction of the interval of integration with to get Since is an even function by assumption, we have for all . Since we then have, So, putting this all together we have, 2. Proof. Similar to part (a) we have, But in this case, since is an odd function, we have for all . Thus, ### One comment

1. Anonymous says:

I think before you come up with the additive property with respect to the interval of integration, you need to point out the integral of f from -b to 0 exists. That means you should write down the contraction / expansion with respect to the interval of integration with k = -1 first, and you can proceed as the rest of the solution.