# Integral of $\int \int_{D} \frac{dxdy}{(4x^2+y^2)^\frac{3}{2}}$

I have to calculate $$\int \int_{D} \frac{dxdy}{(4x^2+y^2)^\frac{3}{2}}$$ where $D = \{(x,y)| 1\leq x \leq 2, |y|\leq \frac{x}{2} \}$

I tried to use polar coordinates but I can't find the limits of integration. Any hint?

First consider the inner integral $$I(x) = \int_{y=-x/2}^{x/2} \dfrac{dy}{(4x^2+y^2)^{3/2}}$$ Setting $y=2x\tan(t)$, we have \begin{align} I(x) & = \int_{t=-\pi/4}^{\pi/4} \dfrac{2x\sec^2(t)dt}{(4x^2\sec^2(t))^{3/2}} = \dfrac1{4x^2} \int_{t=-\pi/4}^{\pi/4}\cos(t)dt = \dfrac{\sqrt2}{4x^2} \end{align} I trust you can finish it off from here.