# Prove the following reduction formula: integrate of ((cosx)^n) dx =1/n(cos^(n-1)x)(sin(x)) + ((n-1)/n) integrate of cos^(n-2)dx

mlehuzzah | Certified Educator

Use integration by parts:

u="cos" ^(n-1) x                              dv = "cos" x

du = (n-1)( "cos"^(n-2) x)( -"sin" x) dx      v="sin" x

Then

int "cos" ^n x dx = "cos"^(n-1) x "sin" x + (n-1) int "cos"^(n-2) x "sin" ^2 x dx

="cos"^(n-1) x "sin" x + (n-1) int "cos"^(n-2) x (1-"cos"^2 x ) dx

="cos"^(n-1) x "sin" x + (n-1) int "cos"^(n-2) x dx - (n-1) int "cos"^n x dx

Notice that we have a int "cos"^n x dx term on both sides of the equation:

int "cos" ^n x dx =[ ...] + [...] - (n-1) int "cos"^n x dx

Adding (n-1) int "cos" ^n x dx to both sides gives:

n int "cos" ^n x dx = "cos" ^(n-1) x "sin" x +(n-1) int "cos" ^(n-2) x dx

Divide by n to get your reduction formula:

int "cos" ^n x dx = (1/n) "cos" ^(n-1) x "sin" x + ((n-1)/n) \int "cos"^(n-2) x dx

mlehuzzah | Certified Educator

http://www.enotes.com/math/q-and-a/prove-following-reduction-formula-integrate-cosx-n-364527