More Calculus Final

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May 1st 2009, 09:29 AM
LanceyPants

More Calculus Final

I'm given a problem and the answer. I get credit for working the steps from beginning to end.

$\int \sin^2 ((\pi/n) x) dx$

Any help would be amazing.
May 1st 2009, 09:46 AM
adhyeta

Quote:

Originally Posted by LanceyPants

I'm given a problem and the answer. I get credit for working the steps from beginning to end.

$\int \sin^2 ((\pi/n) x) dx$

Any help would be amazing.

rewrite the thing as:

http://latex.codecogs.com/gif.latex?...\pi}{n}x%20)dx

try it out now.
(Happy)
May 1st 2009, 09:47 AM
e^(i*pi)

Quote:

Originally Posted by LanceyPants

I'm given a problem and the answer. I get credit for working the steps from beginning to end.

$\int \sin^2 ((\pi/n) x) dx$

Any help would be amazing.

Let $\alpha = \frac{\pi}{n}$ :

$\int \sin^2(\alpha x)$

Remember that $cos(2\alpha x) = 1-2sin^(\alpha x)$ therefore $sin^2(\alpha x) = \frac{1-cos(2\alpha x)}{2} = \frac{1}{2} - \frac{cos(2\alpha x)}{2}$

$I = \frac{1}{2} - \int \frac{cos(2\alpha x)}{2}$

where I is the integral.

Spoiler:

$I = \frac{1}{2} - \int \frac{cos(2\alpha x)}{2}$

$= \frac{1}{2} + \frac{1}{4\alpha}sin(2\alpha x) + C$

Factorising and putting in $\alpha = \frac{\pi}{n}$

$\frac{1}{2}(1+\frac{n}{2\pi}sin(\frac{2\pi x}{n})) + C$

where C is the constant of integration

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