Question

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

Answer 1

See below.

Explanation:

`d/(dx)(cos x sin^(n-1)x)=-sin^nx+(n-1)cos^2xsin^(n-2)x =`
`=-sin^n x+(n-1)(1-sin^2x)sin^(n-2)x =`
`= -sin^nx-(n-1)sin^n x+(n-1)sin^(n-2)x=`
`=-nsin^n x+(n-1)sin^(n-2) x` so calling

`I_n = int sin^n x dx` we have

`I_n-((n-1)/n) I_(n-2)=-1/ncos x sin^(n-1)x` with

`I_0 = x` and
`I_1 = -cos x`

So, with this recurrence equation we can build the response for a given `n`