# integration

• August 7th 2012, 09:54 AM
integration
$\int\cos ^ {2004}x\cos 2004x\ dx$
• August 7th 2012, 11:24 AM
JohnDMalcolm
Re: integration
This can be solved using a bunch of integration by parts with a little help by a useful trig identity.

$\int_{a}^{b} u \frac{dv}{dx} dx = \left. uv \right|_{x=a}^{b} - \int_{a}^{b} \frac{du}{dx} v dx$

Let $u = \cos^{2004}{x} \ , \ \frac{dv}{dx} = \cos{2004x} \Rightarrow v = \frac{1}{2004} \sin{2004x}$

$\int \cos^{2004}{x} \cos{2004x} dx = C + \frac{2004}{2004} \int \sin^{2003}{x} \sin{2004x} dx$

Continue integrating by parts until you notice a pattern. This pattern should take you all the way down to an integrand that you can split up using a handy trig identity. (One that takes the product of two trig functions and gives the sum of two other trig functions).
• August 18th 2012, 12:12 PM