If the fourier integral of a function exists, is it always equal to the fourier cosine integral of that function?

Question

I read that every function which is piecewise continuous and absolutely integrable has a Fourier integral representation which reduces to a Fourier sine or cosine integral accordingly as the function being odd or even. I also read that a function which is neither even nor odd has a Fourier cosine and sine integral representation.

Does that mean if for a function Fourier integral exists, it is always equal to Fourier cosine integral of that function?

Answer 1

A function $f(x)=f_{e}(x)+f_{o}(x)$ where $f_e$ is even, $f_o$ is odd: $$ f_e(x)=\frac{f(x)+f(-x)}{2},\;\; f_o(x)=\frac{f(x)-f(-x)}{2}. $$ If $f$ is even, then $f_e=f$ and $f_o=0$. If $f$ is odd, then $f_e=0$ and $f_o=f$.