Question

If \( (f(x))^2 = 25 + \int_0^x \left[ (f(x))^2 + (f'(x))^2 \right] \, dx \), find the mean of \( f(\ln 1) + f(\ln 2) + \dots + f(\ln 625) \):

• A. 1561
• B. 1675
• C. 1465
• D. 1565

Hint:

When solving integrals involving functions and their derivatives, use differentiation and integration techniques to express the function in a manageable form.

Answer 1

The Correct Option is D

Step 1: Use the given equation.
We are given that: \[ (f(x))^2 = 25 + \int_0^x \left[ (f(x))^2 + (f'(x))^2 \right] \, dx \] Differentiating both sides with respect to \( x \), we get: \[ 2f(x) f'(x) = (f(x))^2 + (f'(x))^2 \] Rearrange this to express in terms of \( f(x) \) and its derivative \( f'(x) \).
Step 2: Solve for \( f(x) \).
Solving the differential equation, we find that: \[ f(x) = 5e^x \]
Step 3: Calculate the sum.
Now, calculate the sum of \( f(\ln 1), f(\ln 2), \dots, f(\ln 625) \): \[ f(\ln n) = 5e^{\ln n} = 5n \] Thus, we need to find: \[ f(\ln 1) + f(\ln 2) + \dots + f(\ln 625) = 5(1 + 2 + 3 + \dots + 625) \] The sum of the first 625 natural numbers is: \[ S = \frac{625(625 + 1)}{2} = \frac{625 \times 626}{2} = 195312.5 \] So, the total sum is: \[ 5 \times 195312.5 = 976562.5 \]
Step 4: Find the Mean.
The mean is the sum divided by the number of terms (625): \[ \text{Mean} = \frac{976562.5}{625} = 1565 \] Final Answer: \[ \boxed{1565} \]