Question

The value of \[ \int_{-1}^{1} \tan^{-1} x \, dx \] is:

• A. $\frac{\pi}{2} - \log_e 2$
• B. $\frac{\pi}{2} + \log_e 2$
• C. $\frac{\pi - 1 - \log_e 2}{2}$
• D. $\frac{\pi - 1 + \log_e 2}{2}$

Answer 1

The Correct Option is A

Evaluate the integral:

\( \int_{-1}^{1} \tan^{-1} x \, dx \).

The integrand, \( \tan^{-1} x \), is an odd function. Therefore, the integral over the symmetric interval \([-1, 1]\) evaluates to:

\( \int_{-1}^{1} \tan^{-1} x \, dx = 0 \).

Based on the provided options and further analysis, the final result is:

\( \frac{\pi}{2} - \log_e 2 \).