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:
\( \frac{\pi}{2} - \log_e 2 \).
List-I (Words) | List-II (Definitions) |
(A) Theocracy | (I) One who keeps drugs for sale and puts up prescriptions |
(B) Megalomania | (II) One who collects and studies objects or artistic works from the distant past |
(C) Apothecary | (III) A government by divine guidance or religious leaders |
(D) Antiquarian | (IV) A morbid delusion of one’s power, importance or godliness |