Question

Evaluate the integral: \[ \int \left[ \frac{1 - \log x}{1 + (\log x)^2} \right]^2 dx \]

• A. \( \frac{1}{1 + (\log x)^2} + C \)
• B. \( \frac{x}{1 + (\log x)^2} + C \)
• C. \( \frac{1}{1 + \log x} + C \)
• D. \( \frac{x}{1 + \log x} + C \)

Hint:

For integrals involving logarithmic functions, consider standard substitutions such as \( u = \log x \), which often simplify the integral.

Answer 1

The Correct Option is B

Step 1: Recognizing the form of the integrand.
The integrand involves the square of a rational function in \( \log x \). This suggests a standard substitution or recognition of the integral's form. Step 2: Solving the integral.
After performing the necessary substitution and simplifying the expression, we find that the integral evaluates to: \[ \frac{x}{1 + (\log x)^2} + C \] Step 3: Conclusion.
Thus, the correct answer is \( \boxed{\frac{x}{1 + (\log x)^2} + C} \).