Mathematics

Integrate the following:
$$\displaystyle \int\limits_{\pi /3}^{\pi /2} {{{(\tan x + \cot x)}^2}dx} $$


SOLUTION
consider, $$I = \displaystyle \int_{\pi/3}^{\pi/2} (\tan x + \cot x)^2dx$$

$$I = \displaystyle \int_{\pi/3}^{\pi/2} (\tan^2 x +2\tan x. \cot x+\cot^2x)dx$$

$$I = \displaystyle \int_{\pi/3}^{\pi/2} (\sec^2-1+2\times 1 + cosec^2 x-1)dx$$

$$I = \displaystyle \int_{\pi/3}^{\pi/2}\sec^2x\, dx + \int_{\pi/3}^{\pi/2}cesec^2x\, dx$$

$$=[\tan x - \cot x]^{\pi/2}_{\pi/3}$$

$$= -\sqrt{3} - \left(-\dfrac{1}{\sqrt{3}}\right) = -\sqrt{3} + \dfrac{1}{\sqrt{3}} = \dfrac{-2\sqrt{3}}{3}$$
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Subjective Medium Published on 17th 09, 2020
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