Question

use integration by parts to prove the following reduction formula:

|cos^nx dx =1/n cos^(n-1)xsinx +(n-1)/n |cos(n-2)xdx

Answer 1

|cos^nx dx = |cos^(n-1)x. cosx dx
Take:
u = cos^(n-1)x
du = (n-1) cos^(n-2)x. (-sinx) dx
dv = cosx dx
v = sinx

By integration by part formula we have:
|cos^nx dx =
= cos^(n-1)x. sinx + (n-1)|sin^2x. cos^(n-2)x dx
= cos^(n-1)x. sinx + (n-1)|(1-cos^2x). cos^(n-2)x dx
= cos^(n-1)x. sinx + (n-1)|cos^(n-2)x dx + (1-n)|cos^nx dx

|cos^nx dx - (1-n)|cos^nx dx = cos^(n-1)x. sinx + (n-1)|cos^(n-2)x dx
n|cos^nx dx = cos^(n-1)x. sinx + (n-1)|cos^(n-2)x dx
|cos^nx dx = (1/n) cos^(n-1)x. sinx + ((n-1)/n)|cos^(n-2)x dx