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Evaluate the definite integral.$\displaystyle \int^{2\pi/3}_{\pi/3} \csc^2 \biggl( \frac{1}{2}t \biggr) \, dt$

$\frac{4 \sqrt{3}}{3}$

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Kristen K.

University of Michigan - Ann Arbor

Samuel H.

University of Nottingham

Michael J.

Idaho State University

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Video Transcript

Okay, We know that you could be substituted as 1/2 teeth, which means that D t is to deal, which means our limits of integration now change from pi over six in the bottom two pi over there in the top. Given what we're substituting. Okay, let's get a new page. Now we can integrate. Let's use the power rule. Which means we increase the experiment by one. And then we divide by the new exponents. In this context, because we have trigger metric functions, we know we can look for the trigger metric shortcut. So we know the integral of Cassie Kit squared of you is negative. Attention. You don't forget the night of science important for the problem. And now we can simply substitute in our bounds. Random before scored of three, divide by three.