SOLUTION: find : int ((cos (x)))^(n) dx from 0 to pi

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Question 1208838: find : int ((cos (x)))^(n) dx from 0 to pi
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!

I won't do this for you, but I'll tell you what I would do.

I would use these two definite integrals:





together with this recursion formula from a table of integrals:

 

Then find 

, , , etc.,

and see if you can find a pattern to the sequence of integrals, and then what
the integrals are approaching as n approaches infinity,



Edwin
Answer by ikleyn(52756)   (Show Source): You can put this solution on YOUR website!
.

I'd like to add my two cents to it . . .


Notice that in the interval  [,],  function  cos(x)  is anti-symmetric relative point  x= .

It implies that all integrals of  cos^n(x)dx  from    to    are zero for all odd positive integer values of  n.



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