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  4. The value of definite integral ∫ limits(7 π/4)(7 π/3) √ tan 2 x d x is equal to

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Q. The value of definite integral $\int\limits_{\frac{7 \pi}{4}}^{\frac{7 \pi}{3}} \sqrt{\tan ^2 x} d x$ is equal to

Integrals

Solution:

Let $I=\int\limits_{\frac{7 \pi}{4}}^{\frac{7 \pi}{3}} \sqrt{\tan ^2 x} d x=\int\limits_{\frac{7 \pi}{4}}^{\frac{7 \pi}{3}}|\tan x| d x=\int\limits_{\frac{7 \pi}{4}}^{2 \pi}-(\tan x) d x+\int\limits_{2 \pi}^{\frac{7 \pi}{3}}(\tan x) d x$
$=(\ln (\cos x))_{315^{\circ}}^{360^{\circ}}-(\ln (\cos x))_{360^{\circ}}^{420^{ n }}=\left(0-\ln \frac{1}{\sqrt{2}}\right)-\left(\ln \frac{1}{2}-0\right. $
$=\frac{-1}{2} \ln \frac{1}{2}-\ln \frac{1}{2}=\frac{-3}{2} \ln \frac{1}{2}=\frac{3}{2} \ln 2 $