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Question Number 34282 by math khazana by abdo last updated on 03/May/18

$$find{\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{dx}{cos\left(x\right)sin\left(x\right)}$$

Commented by math khazana by abdo last updated on 07/May/18

$$I={\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}2\frac{dx}{sin\left(2x\right)}=2{\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{dx}{sin\left(2x\right)}$$$$=2{[\frac{1}{2}ln\mid tanx\mid ]}_{\frac{\pi }{6}}^{\frac{\pi }{3}}=ln\left(tan\left(\frac{\pi }{3}\right)\right)-ln\left(tan\left(\frac{\pi }{6}\right)\right)$$$$=ln\left(\sqrt{3}\right)-ln\left(\frac{1}{\sqrt{3}}\right)=2ln\left(\sqrt{3}\right)=ln\left(3\right).$$

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