I've been struggling with approximating these two integrals
$A=\frac{\delta}{\pi} \int^{\pi/2}_{-\pi/2} \frac{cos^2(k)}{\sqrt {(sin(k)+\tau)^2+\delta^2cos^2(k)}}dk$
$B=\frac{1}{\pi} \int^{\pi/2}_{-\pi/2} sin(k)\frac{sin(k)+\tau}{\sqrt {(sin(k)+\tau)^2+\delta^2cos^2(k)}}dk$
here $0<\delta \ll 1$ and $0<\tau\ll 1$, but the ratio between them is arbitrary.
Since, in the $0< \delta, \tau\ll 1$ case $\frac{1}{\sqrt {(sin(k)+\tau)^2+\delta^2cos^2(k)}}$ has a strong peak at $k=0$, can one expand the whole integrand in the vicinity of $k=0$? For "A" integral, I guess, it is more or less possible, but what about "B"?
There are some logarithmic terms which I am looking for.
Thank you