Question

\[\int\limits_{\pi/3}^{\pi/4} \left( \tan x + \cot x \right)^2 dx\]

Answer 1

\[Let\ I = \int_\frac{\pi}{3}^\frac{\pi}{4} \left( \tan x + \cot x \right)^2 d x . Then, \]
\[I = \int_\frac{\pi}{3}^\frac{\pi}{4} \left( \tan^2 x + \cot^2 x + 2 \tan x \cot x \right) dx\]
\[ \Rightarrow I = \int_\frac{\pi}{3}^\frac{\pi}{4} \left( \tan^2 x + \cot^2 x + 2 \right) dx\]
\[ \Rightarrow I = \int_\frac{\pi}{3}^\frac{\pi}{4} \left( \sec^2 x - 1 + {cosec}^2 x - 1 + 2 \right) dx\]
\[ \Rightarrow I = \int_\frac{\pi}{3}^\frac{\pi}{4} \left( \sec^2 x + {cosec}^2 x \right) dx\]
\[ \Rightarrow I = \left[ \tan x - \cot x \right]_\frac{\pi}{3}^\frac{\pi}{4} \]
\[ \Rightarrow I = \left( 1 - 1 \right) - \left( \sqrt{3} - \frac{1}{\sqrt{3}} \right)\]
\[ \Rightarrow I = \frac{- 2}{\sqrt{3}}\]