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Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Very short answer type question 22

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Answer: \frac{1}{2} \log 2

Hint: you must know the rule of integration for function of x

Given: \int_{0}^{\frac{\pi}{4}} \tan x\; d x

Solution:  \int_{0}^{\frac{\pi}{4}} \tan x\; d x

\begin{aligned} &=\log [\sec x]_{0}^{\frac{\pi}{4}} \\\\ &=\log \left(\sec \frac{\pi}{4}\right)-\log (\sec 0) \\\\ &=\log |\sqrt{2}|-\log |1| \end{aligned}

\begin{aligned} &=\log \sqrt{2}-0 \\\\ &=\log \sqrt{2} \\\\ &=\log (2)^{\frac{1}{2}} \\\\ &=\frac{1}{2} \log 2 \end{aligned}


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