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

Consider the integral:

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

Since \( \tan^{-1} x \) is an odd function, the integral over the symmetric interval \([-1, 1]\) simplifies as follows:

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

Given the expression presented in the options, further consideration and transformations lead to the answer being:

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