Evaluate the following integral : `int_(pi//6)^(pi//3)(t a n x+cot x)^2dx`

Explanation

`int_{frac{pi}{6}}^{frac{pi}{3}}(tan x+cot x)^{2} d x`
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=int_{frac{pi}{6}}^{frac{pi}{3}}(tan ^{2} x+cot ^{2} x+2 tan x cot x) d x`
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=int_{frac{pi}{6}}^{frac{pi}{3}}(sec ^{2} x-1+{cosec}^{2} x-1+2) d x`
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=int_{frac{pi}{6}}^{frac{pi}{3}} sec ^{2} x d x+int_{frac{pi}{6}}^{frac{pi}{3}} {cosec}^{2} x d x`
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=.tan x|_{frac{pi}{6}} ^{frac{pi}{3}}+.(-cot x)|_{frac{pi}{6}} ^{frac{pi}{3}}`
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=(tan frac{pi}{3}-tan frac{pi}{6})-(cot frac{pi}{3}-cot frac{pi}{6})`
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=(sqrt{3}-frac{1}{sqrt{3}})-(frac{1}{sqrt{3}}-sqrt{3})`
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=2 sqrt{3}-frac{2}{sqrt{3}}`
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