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How to evaluate the integral

$$\displaystyle\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over \cos^2x+h^2}dx \int_{0}^{\pi}{\sin x \sin nx \over \cos^2x+h^2}dx$$

and

$$\displaystyle\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over \cos^2x+h_1^2}dx\int_{0}^{\pi}{\sin x \sin nx \over \cos^2x+h_2^2}dx$$

For the first one, I found that the items could be eliminated by the formula $\cos x \cos y={1 \over 2}(\cos(x+y)+\cos(x-y))$ but I have no idea whether such elimination is right or not for $n \rightarrow \infty$. For the second one, I have no idea how to solve it. Note in the second formula $h_1$ and $h_2$ are different.

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  • $\begingroup$ Is there perhaps an error? The first and second ones are the same. In any case, I would say you are on the right track with that identity, and there is a similar one for sine. (To derive it as well as the cosine one you mentioned, combine the angle addition and subtraction identities for sine and cosine) $\endgroup$
    – Eric Auld
    Nov 20, 2013 at 9:50
  • $\begingroup$ Note in the second formula $h_1$ and $h_2$ are different. $\endgroup$
    – sam
    Nov 20, 2013 at 10:45

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