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Math Help - More Calculus Final

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by LanceyPants View Post
    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:



    try it out now.
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  3. #3
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    Quote Originally Posted by LanceyPants View Post
    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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