[tex] b_{n} = \frac{1}{\pi}\int^{\pi}_{-\pi}sin\theta sin n\theta d \theta [/tex](adsbygoogle = window.adsbygoogle || []).push({});

let

[tex] u = sin \theta, \ du = cos \theta d \theta [/tex]

[tex] dv = sin n \theta d \theta, \ v = -\frac{1}{n}cosn \theta [/tex]

[tex] = \left[-\frac{1}{n} sin \theta cos n \theta \right|^{\pi}_{-\pi} + \frac{1}{n} \int^{\pi}_{-\pi} cos \theta cos n \theta d \theta \right] [/tex]

now [tex]-\frac{1}{n} sin \theta cos n \theta \right|^{\pi}_{-\pi} = 0 [/tex]

and

[tex] u = cos\theta, \ du = -sin \theta d\theta [/tex]

[tex] dv = -sin\theta d\theta, \ v = \frac{1}{n}sin n \theta [/tex]

[tex] = \frac{1}{n} \left[\frac{1}{n} cos \theta sin n \theta \right|^{\pi}_{-\pi} + \frac{1}{n} \int^{\pi}_{-\pi} sin \theta sin n\theta \right] [/tex]

I keep getting that to come out to zero but I know it shouldn't. I'm not sure what it should come out to, but it's a Fourier coeff for expanding sin(x). Since sin(x) is an odd function, I know that this coeff should have a value other than 0.

Can someone help me?

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# Integration by parts and coefficients

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