To solve the given integral equation based on integral transformations, we need to analyze the equation:

$\int_{0}^{\infty} f(x) \cos(\lambda x) \, dx = e^{-\lambda}$

This form suggests the use of Fourier or Laplace-related transforms. Let's proceed step by step:

Step 1: Recognize the transform

The left-hand side is a Fourier cosine transform, defined as: $F_c(\lambda) = \int_{0}^{\infty} f(x) \cos(\lambda x) \, dx$ Here, $F_c(\lambda) = e^{-\lambda}$. The goal is to find $f(x)$.

Step 2: Use the inverse cosine transform

The inverse Fourier cosine transform is given by: $f(x) = \frac{2}{\pi} \int_{0}^{\infty} F_c(\lambda) \cos(\lambda x) \, d\lambda$

Substituting $F_c(\lambda) = e^{-\lambda}$, we get: $f(x) = \frac{2}{\pi} \int_{0}^{\infty} e^{-\lambda} \cos(\lambda x) \, d\lambda$

Step 3: Simplify the integral

To evaluate the integral, we use the standard formula: $\int_{0}^{\infty} e^{-a\lambda} \cos(b\lambda) \, d\lambda = \frac{a}{a^2 + b^2}, \quad a > 0$

Here, $a = 1$ and $b = x$. Applying the formula: $\int_{0}^{\infty} e^{-\lambda} \cos(\lambda x) \, d\lambda = \frac{1}{1 + x^2}$

Step 4: Substitute back into the equation

Now substitute the result into the expression for $f(x)$: $f(x) = \frac{2}{\pi} \cdot \frac{1}{1 + x^2}$

Thus, the solution is: $f(x) = \frac{2}{\pi(1 + x^2)}$

Final Answer:

$f(x) = \frac{2}{\pi(1 + x^2)}$

Let me know if you need further details or explanations!

Expanding Knowledge:

1. What is the difference between Fourier cosine and sine transforms?
2. How can the inverse transform be derived?
3. What are the conditions for convergence of these transforms?
4. How does this integral equation relate to Laplace transforms?
5. Can this technique be applied to differential equations?

Tip: Familiarize yourself with key integral transform