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