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JH
Numerade Educator

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Problem 38 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin \theta \cot \theta}{\sec \theta}\ d \theta $

Answer

$\int_{\pi / 6}^{\pi / 3} \frac{\sin \theta \cot \theta}{\sec \theta} d \theta=\frac{\pi}{12}$

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Video Transcript

Let's just start off here by rewriting this integral. So pining for six pi over three, and then we have sign and let me go headed instead of writing one over C camp. Just replace that with co sign and then apply the definition for coach engine that's co sign over sign and then deviate a still. Now let's go ahead and cancel those signs. We have co sign squared and then apply the half angle identity to rewrite Close Science cleared this up here. That's our co sign squared. And now we're using the Half ing Identity, which is replacing co sign squared with one plus co sign to Dana and then all divided by two. And we pulled out the one half of me. So go ahead and integrate this state over to Sines, who data over two times two Plug in those end points tops Piper three. So plug in pi over three first and then minus. And now plug in pi over six. And then here. These values will cancel out there the same, but we're surprising. And then you have pie over six minus pyrrhus. Well, fascist Positive pi over twelve, and that's your final answer