Question

If \[ \int \frac{1}{x(\log x)^2 + 4\log x - 1} \, dx = A \log [ \log x + B ] + K \] where \( K \) is the constant of integration, then \[ \int \frac{1}{x(\log x)^2 + 4\log x - 1} \, dx = A \log [ \log x + C ] + K. \]

• A. \( A = \frac{1}{2\sqrt{5}}, B = (2 - \sqrt{5}), C = (2 + \sqrt{5}) \)
• B. \( A = -\frac{1}{2\sqrt{5}}, B = (2 - \sqrt{5}), C = (2 + \sqrt{5}) \)
• C. \( A = \frac{1}{2\sqrt{5}}, B = (2 + \sqrt{5}), C = (2 - \sqrt{5}) \)
• D. \( A = -\frac{1}{2\sqrt{5}}, B = (2 + \sqrt{5}), C = (2 - \sqrt{5}) \)

Hint:

When dealing with logarithmic integrals, consider substitution to simplify the expression and solve for the constants step by step.

Answer 1

The Correct Option is A

Given the integral and its result as \( A \log [ \log x + B ] + K \), we need to solve for \( A \), \( B \), and \( C \). Step 1: The structure of the integral suggests a substitution involving the logarithmic terms. Step 2: By simplifying and equating terms, we find the values of \( A \), \( B \), and \( C \) to be: \[ A = \frac{1}{2\sqrt{5}}, \quad B = 2 - \sqrt{5}, \quad C = 2 + \sqrt{5}. \] % Final Answer The correct values are \( A = \frac{1}{2\sqrt{5}}, B = (2 - \sqrt{5}), C = (2 + \sqrt{5}) \).