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 $0 \lt m \lt 1$
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 $0 \lt m \lt 1$
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 $O(m^2)$.