Question

How do you find the integral of `int cos^n(x)` if m or n is an integer?

Answer 1

See the explanation for one way to do these.

Explanation:

Even `n` For `n=2k` use
`cos(2x) = 2cos^2x-1` to get the power reduction identity:

`cos^2x = 1/2(1+cos(2x))`

So

`(cos^2x)^k = (1/2(1+cos(2x)))^k`

Expand the power `k` and reduce any

Odd `n`
For `n=2k+1` rewrite as

`cos^(2k+1)x = (cos^2x)^k cosx`

`= (1-sin^2x)^k cosx`

Now use `cos(2x) = 1-2sin^2x` to get the power reduction identity:

`sin^2x = 1/2(1-sin(2x))`.

Us power reduction, expanding the power `k`, then power reduction again and repeat as needed to get an integral of the form
`int(k+a_nsin^(b_n) u + a_(n-1)sin^(b_2) u + * * * +a_1sin^(b_1)u) cosu du`

Then integrate term by term using substitution.