recursive formula

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January 9th 2010, 09:02 PM
alexandrabel90

recursive formula

could someone show me how to derive the recursive formula for
(cos x )^n?

thank you!
January 9th 2010, 09:14 PM
Drexel28

Quote:

Originally Posted by alexandrabel90

could someone show me how to derive the recursive formula for
(cos x )^n?

thank you!

I assume you mean $\int\cos^n(x)\text{ }dx$ . Try using integration by parts.
January 9th 2010, 09:23 PM
alexandrabel90

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..
January 9th 2010, 09:36 PM
Prove It

Quote:

Originally Posted by Drexel28

I assume you mean $\int\cos^n(x)\text{ }dx$ . Try using integration by parts.

$\int{\cos^n{x}\,dx} = \int{\cos^2{x}\cos^{n - 2}{x}\,dx}$

$= \int{(1 - \sin^2{x})\cos^{n - 2}{x}\,dx}$

$= \int{\cos^{n - 2}{x} - \sin^2{x}\cos^{n - 2}{x}\,dx}$

$= \int{\cos^{n - 2}{x}\,dx} - \int{\sin^2{x}\cos^{n - 2}{x}\,dx}$

Use integration by parts on the second integral, letting

$u = \sin{x}$ and $dv = \sin{x}\cos^{n - 2}{x}$ .

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