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...

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

**Moderators:** gmalivuk, Moderators General, Prelates

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

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.)

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

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

^{3}(A)### Re: Which of these methods of integrating cos^3(x) is incorrect?

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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