could someone show me how to derive the recursive formula for
(cos x )^n?
thank you!
yes.
thats what i did. and i kept getting stuck as i was not able to convert the formula to a decreasing function. i kept getting (cos x )^ (n+1) and using 1- sin^2 x = cos^2 x doesnt help..
this is my working:
integrate cos(x) (cos x)^(n-1)
= sin(x) (cos x)^(n-1) +(n-1) integrate sin^2 x (cos x)^(n-1) dx
=sin(x) (cos x)^(n-1) +(n-1) [ - cos ^n x sin x (1/n) + integrate (cos x)^(n+1) (1/n) dx ]
the answer that i'm supoosed to get is
1/n ( cos x)^(n-1) sin x + ((n-1)/n) integrate (cos x )^(n-2) dx
but i cant seem to get that no matter how i try..
$\displaystyle \int{\cos^n{x}\,dx} = \int{\cos^2{x}\cos^{n - 2}{x}\,dx}$
$\displaystyle = \int{(1 - \sin^2{x})\cos^{n - 2}{x}\,dx}$
$\displaystyle = \int{\cos^{n - 2}{x} - \sin^2{x}\cos^{n - 2}{x}\,dx}$
$\displaystyle = \int{\cos^{n - 2}{x}\,dx} - \int{\sin^2{x}\cos^{n - 2}{x}\,dx}$
Use integration by parts on the second integral, letting
$\displaystyle u = \sin{x}$ and $\displaystyle dv = \sin{x}\cos^{n - 2}{x}$.