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**Random Variable** The problem is to show that $\displaystyle \int^{x+ct+L}_{x-ct-L} g(s) \ ds = - \int^{x+ct}_{x-ct} g(s) \ ds $

where g is an odd 2L periodic function and g(L-x)=g(x)

I don't understand the following change of variables: $\displaystyle \int^{x+ct+L}_{x-ct-L} g(s) \ ds = \int^{x+ct}_{x-ct} g(s+L) \ ds $

But if if the above is true, then it's easy to see that $\displaystyle \int^{x+ct}_{x-ct} g(s+L) \ ds = -\int^{x+ct}_{x-ct} g(-s-L) \ ds = - \int^{x+ct}_{x-ct} g(-s+L) \ ds $ $\displaystyle = -\int^{x+ct}_{x-ct} g(s) \ ds $