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Integration (inner product)

  1. Oct 26, 2011 #1
    Inner product:

    [itex]\displaystyle <f,g>=\frac{1}{\pi}\int_{-\pi}^{\pi}fg \ dx=\begin{cases}0 & \ \text{if} \ f=g\\1 & \ \text{if} \ f\neq g\end{cases}[/itex]

    Basis:
    [itex]\displaystyle\left\{\frac{1}{\sqrt{2}},\cos\theta, \sin\theta,\cdots\right\}[/itex]

    I am trying to remember how to integrals of the form:

    [itex]\displaystyle \int_{-\pi}^{\pi}\sin^{a}\theta\cos^b (2\theta) \ d\theta[/itex]

    However, I getting no where.

    I left some guidance with these two integrals and I should be good to go then.

    [itex]\displaystyle\int_{-\pi}^{\pi}\sin^4\theta \ d\theta[/itex]

    [tex]\Rightarrow\int_{-\pi}^{\pi}\left(\frac{1}{2}-\frac{\cos(2\theta)}{2}\right)^2 \ d\theta[/tex]

    [tex]\Rightarrow \int_{-\pi}^{\pi}\left(\left(\frac{1}{\sqrt{2}}\right)^2-\frac{\cos(2\theta)}{2}\right)^2 \ d\theta[/tex]

    Now, I am drawing a blank.

    The other one I need guidance on is:

    [itex]\displaystyle\int_{-\pi}^{\pi}\sin^4\theta\cos(2\theta) \ d\theta[/itex]

    [tex]\Rightarrow\int_{-\pi}^{\pi}\left(\left(\frac{1}{\sqrt{2}}\right)^2-\frac{\cos(2\theta)}{2}\right)^2\cos(2\theta) \ d\theta[/tex]

    [tex]\Rightarrow\int_{-\pi}^{\pi}\frac{\cos(4\theta)\cos(2\theta)}{4} \ d\theta[/tex]

    Now I am stuck again.
     
    Last edited: Oct 26, 2011
  2. jcsd
  3. Oct 27, 2011 #2
    Bad math:

    [tex]\int_{-\pi}^{\pi}\sin^4\theta\cos(2\theta) \ d\theta=[/tex]
    [tex]\int_{-\pi}^{\pi}\left[\frac{\cos(2\theta)}{4}-\frac{\cos^2(2\theta)}{2}+\frac{\cos(2\theta)* \cos^2(2\theta)}{4}\right] \ d\theta=[/tex]
    [tex]\int_{-\pi}^{\pi}\left[\frac{\cos(2\theta)}{4}-\left(\frac{1}{4}+\frac{\cos(4\theta)}{4}\right)+\frac{\cos(2\theta)}{8}+\frac{\cos(2\theta)*\cos(4\theta)}{8}\right] \ d\theta[/tex]
    (the Latex is correct so I don't know why it is all jacked up)

    [tex]\int_{-\pi}^{\pi}\left[\frac{3\cos(2\theta)}{8}-\frac{1}{4}-\frac{\cos(4\theta)}{4}+\frac{\cos(2\theta)*\cos(4\theta)}{8}\right] \ d\theta[/tex]
     
  4. Oct 31, 2011 #3
    Can anyone provide any guidance?
     
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