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JH
Numerade Educator

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Problem 7 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{2}} \cos^2 \theta d \theta $

Answer

$\int_{0}^{\pi / 2} \cos ^{2} \theta d \theta=\frac{\pi}{4}$

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Video Transcript

this problem is from Chapter seven, Section two. Problem number seven in the book Calculus Early Transcendental Sze, eighth Edition. My James Dorf. So here we have a definite integral from zero pie over two of co sign squared of theta D data. So since there's no signs in this problem, it's only a co sign, and it's an even power. It's going to be useful to use this trick to metric identity for co science, where SOCO Sign Square is one plus co sign of two data, all divided by two. So we simply replaced co sign squared with this other expression tohave integral from zero to Piper, too. And then we have a one half plus co sign to date over, too. So now let's integrate these from separately. So one half the integral that just becomes data over to and for the other term, this becomes science with NATO over four, and this is coming from the use up u equals to data, and we're evaluating this new expression zero and powerful, too. So it's played by over to first for the so we have a power for plus sign two times Pi over to is just pie. So sign a pie over four and now we plug it zero as well. When we plug in zero for data, we have a zero for two plus sign of zero over four. So using our knowledge from the unit Circle and trick, we know that Santa Pie a zero sign of zero zero. Sorry. Final answer paper for