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प्रश्न
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उत्तर
\[\text{Let, }I = \int_0^\frac{\pi}{2} \log \tan x\ dx ...................(1)\]
\[ = \int_0^\frac{\pi}{2} \log \tan\left( \frac{\pi}{2} - x \right) dx ....................\left[ Using, \int_0^a f\left( x \right) dx = \int_0^a f\left( a - x \right) dx \right]\]
\[ = \int_0^\frac{\pi}{2} \log cot x\ dx ....................(2)\]
\[\text{Adding (1) and (2) we get}\]
\[2I = \int_0^\frac{\pi}{2} \log \tan x d x + \int_0^\frac{\pi}{2} \log cotx\ dx\]
\[ = \int_0^\frac{\pi}{2} \log\left( \tan x \times cotx \right)dx\]
\[ = \int_0^\frac{\pi}{2} \log1 dx = 0\]
\[\text{Hence, }I = 0\]
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