Problem 3 ( 20 points). Find the Fourier cosine transform of $f(x)=e^{-x}+2 e^{-2 x}$ without usi the Table. Note: $\int_{0}^{\infty}\left(e^{a x}\right) \cos w x d x=\frac{e^{a x}(a \cos (w x)+w \sin (w x))}{a^{2}+w^{2}}$

Step 1 ：Define the function \(f(x) = e^{-x} + 2e^{-2x}\).

Step 2 ：Define the Fourier cosine transform formula as \(F(w) = \sqrt{\frac{2}{\pi}} \int_{0}^{\infty} f(x) \cos(wx) dx\).

Step 3 ：Substitute the function \(f(x)\) into the Fourier cosine transform formula.

Step 4 ：Simplify the result to get the Fourier cosine transform of the function \(f(x)\).

Step 5 ：The Fourier cosine transform of the function \(f(x) = e^{-x} + 2e^{-2x}\) is given by the expression \(F(w) = \frac{\sqrt{2}(5w^2 + 8)}{\sqrt{\pi}(w^2 + 1)(w^2 + 4)}\) for \(w \geq 0\).

Step 6 ：Final Answer: \(\boxed{F(w) = \frac{\sqrt{2}(5w^2 + 8)}{\sqrt{\pi}(w^2 + 1)(w^2 + 4)}}\) for \(w \geq 0\).