pi-6-pi-3-1-2-cot-2-2-d-

Question Number 21707 by Isse last updated on 01/Oct/17

\int_{π / 6}^{π / 3} \frac{1}{2} {c o t}^{2} 2 θ d θ

Commented by Tikufly last updated on 01/Oct/17

I = \frac{1}{2} \int_{π / 6}^{π / 3} ({cosec}^{2} 2 θ - 1) d θ

= \frac{1}{2} \int_{π / 6}^{π / 3} {cosec}^{2} 2 θ - \frac{1}{2} \int_{π / 6}^{π / 3} 1 d θ

= - \frac{1}{4} {[\cot 2 θ]}_{π / 6}^{π / 3} - \frac{1}{2} {[θ]}_{π / 6}^{π / 3}

= - \frac{1}{4} {\cot (\frac{2 θ}{3}) - \cot (\frac{θ}{3})} - \frac{1}{2} (\frac{π}{3} - \frac{π}{6})

= - \frac{1}{4} (- \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{3}}) - \frac{1}{2} (\frac{π}{6})

= \frac{1}{2 \sqrt{3}} - \frac{π}{12}

Commented by Isse last updated on 01/Oct/17

t h a n k s

Commented by Tikufly last updated on 02/Oct/17

Welcome

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