∫^π/2_π/3(tan x+ x)^2dx =∫^π/2_π/3(sin^2x+cos^2 xsin xcos x)^2dx =∫^π/2_π/3(4sin^22 x)^2dx = 2[ 2x]^π/2_π/3 = 2√(3) ...

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Standard Values of Trigonometric Ratios

∫π/2π/3tan x+...

Question

$\int_{π / 3}^{π / 2} (tan x + cot x)^{2} d x$ .

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Solution

$\int_{π / 3}^{π / 2} (tan x + cot x)^{2} d x$
$= \int_{π / 3}^{π / 2} (\frac{{sin}^{2} x + {cos}^{2} x}{sin x cos x})^{2} d x$
$= \int_{π / 3}^{π / 2} (\frac{4}{{sin}^{2} 2 x})^{2} d x$
$= - 2 [cot 2 x]_{π / 3}^{π / 2}$
$= \frac{- 2}{\sqrt{3}}$

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