Which of these methods of integrating cos^3(x) is incorrect?

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Patashu
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Which of these methods of integrating cos^3(x) is incorrect?

Postby Patashu » Tue May 27, 2008 6:11 am UTC

cos^3(x)
= cos^2(x)cos(x)
= (1-sin^2(x))cos(x) [by pythagorean identity]
= cos(x)-sin^2(x)cos(x) [by distributive property]
Let u = sin(x) such that du/dx = cos(x) and thus sin^2(x)cos(x)dx = u^2du = u^3/3 = sin^3(x)/3

Integrated = sin(x) - sin^3(x)/3

cos^3(x)
= cos^2(x)cos(x)
= 1/2(cos(x))(1+cos(2x)) [double angle result]
= 1/2cos(x) + 1/2cos(x)cos(2x) [by distributive property]
= 1/2cos(x) + 1/4(cos(2x+x) + cos(2x-x)) [angle addition result as cos(a+b) + cos(a-b) = cosacosb + sinasinb + cosacosb - sinasinb = 2cosacosb]
= 3/4cos(x) + 1/4cos(3x)

Integrated = 3sin(x)/4 + sin(3x)/12

One of these things just doesn't belong...

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jaap
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Re: Which of these methods of integrating cos^3(x) is incorrect?

Postby jaap » Tue May 27, 2008 6:21 am UTC

Patashu wrote:Integrated = sin(x) - sin^3(x)/3
Integrated = 3sin(x)/4 + sin(3x)/12

One of these things just doesn't belong...


Don't you think they might be equal?
(They could differ of course by a constant of integration, but plugging in x=0 shows that constant would have to be 0.)

masher
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Re: Which of these methods of integrating cos^3(x) is incorrect?

Postby masher » Tue May 27, 2008 6:44 am UTC

Identity: sin(3A) = 3sin(a) - 4sin3(A)

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Patashu
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Re: Which of these methods of integrating cos^3(x) is incorrect?

Postby Patashu » Tue May 27, 2008 6:45 am UTC

jaap wrote:
Patashu wrote:Integrated = sin(x) - sin^3(x)/3
Integrated = 3sin(x)/4 + sin(3x)/12

One of these things just doesn't belong...


Don't you think they might be equal?
(They could differ of course by a constant of integration, but plugging in x=0 shows that constant would have to be 0.)


:?

I swear I checked them in winplot while I was at school and they were different, but I just plugged them in again and they ARE the same.
Either it's a bug in a newer version or I fail at typing.


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