Question

\(\int \frac{4 \tan^{-1} x}{1 + x^2} \, dx \)

• A. \( \frac{\pi^2}{4} \)
• B. \( \frac{\pi^2}{8} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{8} \)

Hint:

When encountering integrals involving inverse trigonometric functions, remember to use the derivative relationships and identities for simplification.

Answer 1

The Correct Option is B

We are given: \[ I = \int \frac{4 \tan^{-1} x}{1 + x^2} \, dx \] Recognize that the derivative of \( \tan^{-1} x \) is \( \frac{1}{1 + x^2} \), which simplifies the integral. Therefore, we can rewrite the integral as: \[ I = 4 \int \tan^{-1} x \, d(\tan^{-1} x) \] This is a standard integral whose result is \( \frac{\pi^2}{8} \). Thus, the correct answer is \( \frac{\pi^2}{8} \).