Solution of Integral equations (using Fourier Transform)

Question

Please help me to solve the Integral equation :

$$\int_0^\infty f(x) \cos sx \mathrm{d}x = e^{-s}$$

using Fourier Transform $s>0$

Answer 1

Hint:

First see that the equation make sense if the left ahnd side is also and even function that is we have, since the right hand side is an even function too. $$e^{-|s|} =\int_0^\infty f(x) \cos sx \mathrm{d}x = \frac12\int_0^\infty f(x)e^{isx}dx +\frac12\int_0^\infty f(x)e^{-isx}dx \\ = \frac12\int_{-\infty}^0 f(-x)e^{-isx}dx +\frac12\int_0^\infty f(x)e^{-isx}dx = \color{red}{\int_\Bbb R \widetilde{f}(x)e^{-isx}dx }$$

where $$ \widetilde{f}(x) =\frac{f(x)+f(-x)}{2}$$ Using the fourier inverse we get

$$ \widetilde{f}(x) =\color{red}{\frac{1}{2\pi}\int_\Bbb R e^{-|s|} e^{isx}ds=.......}$$