If f $:\mathrm{R}\to \mathrm{R}\text{ and }\mathrm{g}:\mathrm{R}\to \mathrm{R}$  are one to one, real valued functions, then the value of the integral ${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }} \left[\mathrm{f}\right(\mathrm{x})+\mathrm{f}(-\mathrm{x}\left)\right]\left[\mathrm{g}\right(\mathrm{x})-\mathrm{g}(-\mathrm{x}\left)\right]\mathrm{dx} \mathrm{is}$

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