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Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Very short answer type question 14

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Answer: 0

Hint: You must know the integration rules of trigonometric function with its limits

Given: \int_{0}^{\frac{\pi}{2}} \log \tan x \; d x

Solution:  \int_{0}^{\frac{\pi}{2}} \log \tan x \; d x            .................(i)

\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \log \tan \left(\frac{\pi}{2}-x\right) d x

    =\int_{0}^{\frac{\pi}{2}} \log \cot x \; d x            ..................(ii)

Adding (i) and (ii)

2 I=\int_{0}^{\frac{\pi}{2}} \log (\tan x) d x+\int_{0}^{\frac{\pi}{2}} \log (\cot x) d x

    =\int_{0}^{\frac{\pi}{2}}(\log \tan x+\log \cot x) d x

    =\int_{0}^{\frac{\pi}{2}}(\log \tan x \cdot \cot x) d x \quad[\because \log m+\log n=\log m n]

    =\int_{0}^{\frac{\pi}{2}}(\log 1) d x \quad[\because \tan x \cdot \cot x=1]

    \begin{aligned} &=\int_{0}^{\frac{\pi}{2}}(0) d x \quad[\because \log 1=0] \\\\ &=0 \end{aligned}



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