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Math Help - Fourier Series Integration Fourmula

  1. #1
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    Question Fourier Series Integration Fourmula

    This is one of the example from the book named Higher Engg. Mathematics III by Dr. Kachot
    Obtain the fourier series for:
    f(x)=\frac{1}{4} (\pi-x)^2 ; 0\leq x \leq 2\pi \\=f(x+2\pi);otherwise

    Here the value of
    a_{0}=\frac{1}{\pi}\int_{0}^{2\pi} f(x)dx = \frac{1}{\pi}\int_{0}^{2\pi} \frac{(\pi-x)^2}{4}dx = \frac{1}{4\pi}[-\frac{(\pi-x)^3}{3}]_{0}^{2\pi}

    Please explain the above step. Didn't get which integration formula the author applied. Also the minus sign is bugging me. Where did it come from?
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  2. #2
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    Re: Fourier Series Integration Fourmula

    Minus Sign - Think Chain Rule.

    \int x^2\;dx = \frac{x^{3}}{3} + C
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  3. #3
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    Unhappy Re: Fourier Series Integration Fourmula

    Chain rule? Didn't get you
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  4. #4
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    Re: Fourier Series Integration Fourmula

    Quote Originally Posted by maulin View Post
    Chain rule? Didn't get you
    Make the substitution

    u=\pi-x

    \frac{du}{dx}=-1\Rightarrow\ dx=-du

    Then the integral becomes

    -\frac{1}{4{\pi}}\int_{x=0}^{x=2{\pi}}u^2du
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  5. #5
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    Re: Fourier Series Integration Fourmula

    Quote Originally Posted by maulin View Post
    Chain rule? Didn't get you
    \frac{d}{dx}f(x) = f'(x)

    \frac{d}{dx}f(-x) = -f'(-x)
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  6. #6
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    Re: Fourier Series Integration Fourmula

    Thanks, now I got it!
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