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Can you help me to show that

$\int^{\pi/2}_{0}{d\theta\over (1-m^2\cos^2\theta)^2} \approx {(2-m^2)\pi\over4(1-m^2)^{3/2}}$

to first order, such that $0 \lt m \lt 1$

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What do you mean, "to first order"? Do you mean, to $O(m^2)$ on either side? If so, then the LHS becomes

$\int_{0}^{\frac{\pi}{2}} d{\theta} (1+2 m^2 \cos^2 {\theta})$ which evaluates to $\frac{\pi}{2} (1+m^2)$. The RHS takes precisely this value upon a Taylor expansion to $O(m^2)$.

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    Sort of. Still, I think you are not formulating the problem correctly, I think. It seems to me that you are trying to find a simple, analytical expression for the original integral for all values of m that works within some error bound. Adding the epsilon in the denominator isn't going to help in this case. What will help is an expansion about m=0, and an expansion about 1/m = 0, and then a combination of the two.2012-12-19