How to compute the following integral using elementary methods (high school methods).

\begin{equation}\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\qquad;\qquad\text{for}\, \alpha>0\end{equation}

Honestly, I don't know how to compute this integral. I have posted this problem in other forum and I only got a link direction to another problem but it didn't help me so that's why I post the problem here. So far I could manage to get \begin{equation} \frac{e^{ix}}{x-\alpha e^{ix}}=\frac{x\cos x-\alpha}{x^2-2\alpha x\cos x+\alpha^2}+i\frac{x\sin x}{x^2-2\alpha x\cos x+\alpha^2} \end{equation} or \begin{equation} \frac{e^{ix}}{x-\alpha e^{ix}}=\frac{1}{\alpha(\beta xe^{-ix}-1)}\qquad;\qquad\text{where}\, \beta=\frac{1}{\alpha} \end{equation} but none of them is easy to be computed. These are related questions that might help: [1] and [2]. Any help would be greatly appreciated. Thank you.

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    $\begingroup$ I don't think there is a closed formula for the primitive. You will need to resort to methods for definite integrals, which I wouldn't qualify as elementary. $\endgroup$ – Yves Daoust Jun 13 '14 at 7:55
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    $\begingroup$ There is no elementary methods for this problem. I even wonder if the antiderivatives will be available. $\endgroup$ – Claude Leibovici Jun 13 '14 at 8:59
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    $\begingroup$ :| High school methods? Can we update to... say... more than high school methods? $\endgroup$ – Simply Beautiful Art Jan 5 '17 at 22:11

Inserting $k$ parameter to the problem :

$$f(k) \overset{\mathrm{def}}{=} \int_{0}^{\pi}\frac{e^{ikx}}{x-\alpha e^{ix}}$$

Where the original integral $I$ is the same as $f(1)$ i.e. $$I=f(0)$$

Differentiating with respect to k assuming that we can do that :

$$f'(k) = \int_{0}^{\pi}\frac{ix e^{ikx}}{x-\alpha e^{ix}} $$

See that f(k) solves delay differential equation

$$f'(k)-i\alpha f(k+1) = \int_{0}^{\pi}\frac{i(x - \alpha e^{ix}) e^{ikx}}{x-\alpha e^{ix}} = \frac{1}{k}\left(e^{ik\pi}-1\right) $$

Taking the limit as $k\to 0$

$$f'(0)-i\alpha I = i\pi $$

Then maybe you may proceed further...


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