Question

If for non-zero \(x\), \( cf(x) + df\left(\frac{1}{x}\right) = |\log |x|| + 3 \), where \( c \neq d \), then \( \int_1^c f(x)\,dx = \):

• A. \( \frac{(c - d)(2e - 1)}{c^2 - d^2} \)
• B. \( \frac{(c - d)(3e - 2)}{c^2 - d^2} \)
• C. \( \frac{(c - d)(3e + 2)}{c^2 - d^2} \)
• D. \( \frac{(c - d)(2e + 1)}{c^2 - d^2} \)

Hint:

To solve differential equations involving both \( f(x) \) and \( f\left(\frac{1}{x}\right) \), use substitution and properties of logarithmic functions to simplify the expression.

Answer 1

The Correct Option is B

Step 1: Analyze the given equation. We are given: \[ cf(x) + df\left(\frac{1}{x}\right) = |\log |x|| + 3 \] This equation contains both \( f(x) \) and \( f\left(\frac{1}{x}\right) \), so we will proceed to solve it by examining the behavior of \( f(x) \). 
Step 2: Consider a substitution for \( f(x) \). Assume \( f(x) = g(x) \), where \( g(x) \) is a function to be determined. First, try substituting this function into the equation. We already know that the equation must hold true for \( x \) in the given domain, and this will allow us to solve for \( f(x) \). 
Step 3: Set up the integral expression. Now, we need to compute the definite integral of \( f(x) \) from 1 to \( c \), which is given by: \[ \int_1^c f(x)\, dx \] We will now proceed with integrating the expression using the appropriate methods, possibly simplifying using known integration techniques.