Question

How do you use integration by parts to establish the reduction formula `intcos^n(x) dx = (1/n)cos^(n-1)(x)sin(x)+(n-1)/nintcos^(n-2)(x)dx` ?

Answer 1

Let
`I=int cos^nxdx`.

Let `u=cos^{n-1}x` and `dv=cosxdx`
`Rightarrow du=-(n-1)cos^{n-2}x sinxdx` and `v=sinx`

By Integration by Parts,
`I=cos^{n-1}x sinx+(n-1)int cos^{n-2}xsin^2xdx`
By `sin^2x=1-cos^2x`,
`I=cos^{n-1}x sinx+(n-1)int cos^{n-2}x(1-cos^2x)dx`
By splitting the last integral,
`I=cos^{n-1}x sinx+(n-1)int cos^{n-2}xdx-(n-1)I`
By adding `(n-1)I`,
`nI=cos^{n-1}x sinx+(n-1)int cos^{n-2}xdx`
By dividing by `n`,
`I=1/ncos^{n-1}xsinx+{n-1}/nintcos^{n-2}xdx`