I did the case $p=1$ here. The generalization to higher $p$ may involve higherorder derivatives as follows: $$\begin{align}K_p &= \int_0^{\pi/2} dx \frac{x^{2 p}}{1+\cos^2{x}} = \frac1{2^{4 p1}} \int_{\pi}^{\pi} dy \frac{y^{2 p}}{3+\cos{y}} \end{align}$$ So define, as before, $$J(a) = \int_{\pi}^{\pi} dy \frac{e^{i a y}}{3+\cos{y}} $$ Then $$K_p = \frac{(1)^p}{2^{4 p1}} \left [\frac{d^{2 p}}{da^{2 p}} J(a) \right ]_{a=0}$$ […]

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