To find $f(x)$ using the Fourier integral of cosine, we typically use the Fourier cosine transform. Here's how it works step by step:

1. Fourier Cosine Transform

The Fourier cosine transform of a function $f(x)$ is given by:

$F_c(\omega) = \int_{0}^{\infty} f(x) \cos(\omega x) \, dx$

This transform expresses $f(x)$ in terms of cosine functions weighted by a frequency-dependent factor $F_c(\omega)$.

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2. Inverse Fourier Cosine Transform

Once you have $F_c(\omega)$, the original function $f(x)$ can be reconstructed using the inverse Fourier cosine transform:

$f(x) = \frac{2}{\pi} \int_{0}^{\infty} F_c(\omega) \cos(\omega x) \, d\omega$

This formula reconstructs the function $f(x)$ from its Fourier cosine transform.

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3. Steps to Solve

Here’s how you can solve for $f(x)$:

(a) Compute $F_c(\omega)$:

Evaluate the Fourier cosine transform of the function $f(x)$ using the given formula: $F_c(\omega) = \int_{0}^{\infty} f(x) \cos(\omega x) \, dx$

(b) Reconstruct $f(x)$:

Substitute $F_c(\omega)$ into the inverse formula to find $f(x)$: $f(x) = \frac{2}{\pi} \int_{0}^{\infty} \left( \int_{0}^{\infty} f(t) \cos(\omega t) \, dt \right) \cos(\omega x) \, d\omega$

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Example

If $f(x) = e^{-ax}$ for $a > 0$, we compute:

1. Fourier Cosine Transform: $F_c(\omega) = \int_{0}^{\infty} e^{-ax} \cos(\omega x) \, dx$ This integral evaluates to: $F_c(\omega) = \frac{a}{a^2 + \omega^2}$

2. Inverse Fourier Transform: $f(x) = \frac{2}{\pi} \int_{0}^{\infty} \frac{a}{a^2 + \omega^2} \cos(\omega x) \, d\omega$ This integral reconstructs $f(x) = e^{-ax}$.

Would you like to dive deeper into an example, or have any specific part clarified?

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Questions for Further Exploration:

1. How does the Fourier cosine transform differ from the sine transform?
2. What conditions must $f(x)$ satisfy for its Fourier integral to exist?
3. Can you derive the Fourier cosine transform for $f(x) = \sin(ax)$?
4. How is the Fourier cosine transform related to the Laplace transform?
5. What are some practical applications of Fourier cosine transforms?

Tip:

When working with Fourier integrals, pay close attention to the symmetry properties of $f(x)$. Cosine transforms are especially useful for even functions.