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January 21, 2017
Posted by **Sean** on Friday, March 14, 2008 at 5:55pm.

Integral of (cos(x\8))^3, defined from

-(4 x pi)/3) to (4 x pi)/3)

- Calculus - Anti-Derivatives -
**Anonymous**, Friday, March 14, 2008 at 6:22pmcos^3(x)dx = cos^2(x)cos(x)dx =

[1-sin^2(x)]cos(x)dx =

[1-sin^2(x)]d(sin(x)) - Calculus - Anti-Derivatives -
**drwls**, Friday, March 14, 2008 at 6:27pmIs cos(x)8 supposed to be

cos 8x, (cosx)^8 or 8 cosx? - Calculus - Anti-Derivatives -
**Sean**, Friday, March 14, 2008 at 6:29pmIts actually supposed to be (cos(x/8))^3.

With what Anonymous answered, I see he left out the x/8, instead only using x. Would you, Drwls, mind using x/8? - Calculus - Anti-Derivatives -
**drwls**, Friday, March 14, 2008 at 6:52pmSorry, I missed the \ mark. It is always better to use / for fractions when typing.

Without messing with trig identities for cos (x/8), let's just substitute u for x/8. Then your integral becomes

Integral of (cos u)^3

= (1- sin^2 u) cos u du , from

u = -(pi/6) to (pi/6)

Now let sin u = v

dv = cos u du

and your integral becomes

Integral of (1- v^2)dv ,

from v = -1/2 to 1/2,

since that is what v is when u = +/- pi/6

Now it should be easy!

The indefinite integral is v - v^3/3. Evaluate it at v = 1/2 and -1/2.

My answer: 11/24 - (-11/24) = 11/12

Don't trust my algebra. Check it yourself. - Calculus - Anti-Derivatives -
**Sean**, Friday, March 14, 2008 at 8:09pmI can follow everything except when you change the (4 x pi)/3 to pi/6, and change pi/6 into 1/2. Could you clear that up for me?

- Calculus - Anti-Derivatives -
**Damon**, Friday, March 14, 2008 at 8:19pmthe upper limit for example was

x = 4 pi/3

but

u = x/8

so the upper limit using u instead of x is

u = (4 pi/3) /8

or

u = pi/6

then it changes again to go from u to v

v = sin u

so at that same upper limit where x = 4 pi/3 and u = pi/6, we need to find v

so

v at this upper limit is:

v = sin (pi/6) = sin 30 degrees = 1/2 - Calculus - Anti-Derivatives -
**Sean**, Friday, March 14, 2008 at 8:38pmOh. Thank you, Damon, and Drwls. That really helped me out.