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Problem 11 Medium Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^\pi \cos^6 \theta\ d \theta $

Answer

$\frac{5 \pi}{16}$

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Video Transcript

Okay. This question has its evaluating an integral co sign to the sixth from zero pie. So to do this, we're gonna have to use the integral for co sign to the end power, which you can see here. So just applying this formula since all we have is theta we get co sign of well, and his six. So we get five with data sign fada over and which is six plus and minus one over end. So n minus one is five, and and it's six times the integral. And I'm just gonna leave the limits offer known well, plugging at the very end of co sign to the n minus two, which is for Okay, So now we have the Inter Grove Coast under the fourth. So we're gonna apply the same formula again. So we get just copying everything down again. Okay, so we get coastline to the fourth, so and is now four. So we get co sign cubed data sign of theta over four plus and minus one is three and is for times the integral of co sign squared fada d theta. Now we apply the Formula One last time or you could just evaluate the inner girl off co sign squared, which is one that you may have memorized You may not. Doesn't matter either way, there's just a lot of stuff to write down. Okay, and then the integral of co sign squared is 1/2 fada plus sign of tooth Ada over four. And we're evaluating this whole thing from zero to pie so you can verify this, but just by inspecting it, we're going to see that the lower them it is zero, because every term has a science data or a regular theater attached somewhere and sign of 00 So we only have to consider the upper limit. And since every term has assigned data wouldn't re plug in pie sign of pie a zero, so we only have to consider terms without a sign in it. So that leaves one term. So this is equal to zero plus 56 times zero plus 3/4 times, 1/2 times pi plus zero, and this gives us a final answer of well, 56 times. 3/4 is 15 over 24 times pi over, too, and that's equal to 15 pie over 48 and then that could be simplify it even more to five pi over 16 as our final answer. So again, things simplified really nicely because the scientist I made a bunch of things cancel. So basically, what happened was this this one away, this one away and this one away. So all we're left with was just the fate of

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