Find the Fourier Cosine Transform of𝑓(𝑥) = 𝑥 𝑒−5𝑥.

To find the Fourier Cosine Transform of 𝑓(𝑥) = 𝑥 𝑒−5𝑥, we can follow these steps:

Step 1: Determine the interval of integration. Since we are dealing with the Fourier Cosine Transform, the interval of integration is from 0 to infinity.

Step 2: Write the Fourier Cosine Transform integral. The Fourier Cosine Transform of 𝑓(𝑥) is given by the integral of 𝑓(𝑥) multiplied by cosine(𝑛𝜋𝑥/𝐿), where 𝑛 is an integer and 𝐿 is the length of the interval of integration. In this case, 𝐿 is infinity, so we can write the integral as:

𝐹(𝑛) = ∫[0,∞] 𝑓(𝑥)cos(𝑛𝜋𝑥/𝐿) 𝑑𝑥

Step 3: Substitute the given function 𝑓(𝑥). Substituting 𝑓(𝑥) = 𝑥 𝑒−5𝑥 into the integral, we have:

𝐹(𝑛) = ∫[0,∞] 𝑥 𝑒−5𝑥 cos(𝑛𝜋𝑥/𝐿) 𝑑𝑥

Step 4: Simplify the integral. To simplify the integral, we can use integration by parts. Let's define 𝑢 = 𝑥 and 𝑑𝑣 = 𝑒−5𝑥 cos(𝑛𝜋𝑥/𝐿) 𝑑𝑥. Then, 𝑑𝑢 = 𝑑𝑥 and 𝑣 = −(1/5)𝑒−5𝑥 sin(𝑛𝜋𝑥/𝐿).

Using the integration by parts formula, the integral becomes:

𝐹(𝑛) = 𝑢𝑣∣[0,∞] - ∫[0,∞] 𝑣𝑑𝑢

Simplifying further, we have:

𝐹(𝑛) = −(1/5)𝑥𝑒−5𝑥 sin(𝑛𝜋𝑥/𝐿)∣[0,∞] + (1/5)∫[0,∞] 𝑒−5𝑥 sin(𝑛𝜋𝑥/𝐿) 𝑑𝑥

Step 5: Evaluate the integral. To evaluate the remaining integral, we can use the Fourier Sine Transform formula. The Fourier Sine Transform of 𝑔(𝑥) = 𝑒−𝑎𝑥 sin(𝑏𝑥) is given by:

𝐺(𝑏) = ∫[0,∞] 𝑔(𝑥) sin(𝑏𝑥) 𝑑𝑥

In this case, 𝑔(𝑥) = 𝑒−5𝑥 sin(𝑛𝜋𝑥/𝐿), so we can write the integral as:

𝐺(𝑛𝜋/𝐿) = ∫[0,∞] 𝑒−5𝑥 sin(𝑛𝜋𝑥/𝐿) 𝑑𝑥

Using the Fourier Sine Transform formula, we find that 𝐺(𝑛𝜋/𝐿) = (𝑛𝜋/𝐿) / (𝑛^2𝜋^2/𝐿^2 + 25).

Step 6: Substitute the result back into the previous integral. Substituting 𝐺(𝑛𝜋/𝐿) = (𝑛𝜋/𝐿) / (𝑛^2𝜋^2/𝐿^2 + 25) into the previous integral, we have:

𝐹(𝑛) = −(1/5)𝑥𝑒−5𝑥 sin(𝑛𝜋𝑥/𝐿)∣[0,∞] + (1/5)𝐺(𝑛𝜋/𝐿)

Since the interval of integration is from 0 to infinity, the first term 𝑥𝑒−5𝑥 sin(𝑛𝜋𝑥/𝐿) evaluated at infinity will be zero. Therefore, we are left with:

𝐹(𝑛) = (1/5)𝐺(𝑛𝜋/𝐿)

Substituting 𝐺(𝑛𝜋/𝐿) = (𝑛𝜋/𝐿) / (𝑛^2𝜋^2/𝐿^2 + 25), we finally get:

𝐹(𝑛) = (1/5) * (𝑛𝜋/𝐿) / (𝑛^2𝜋^2/𝐿^2 + 25)

This is the Fourier Cosine Transform of 𝑓(𝑥) = 𝑥 𝑒−5𝑥.