# Definite integral $\int_0^{2\pi}\frac{ab}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta))}}\cos^2(\theta) d\theta$

Could you help me finding the following definite integral, with $a$ and $b$ constants? Thank you! $$\int_0^{2\pi}\frac{ab}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta))}}\cos^2(\theta) d\theta$$

• A good way to analyse such integrals is to give specific values for a,b and then see how in general it could be evaluated. – Torsten Hĕrculĕ Cärlemän Dec 22 '13 at 17:05
• – Lucian Dec 22 '13 at 17:47
• Try this technique. – Mhenni Benghorbal Dec 22 '13 at 19:16

Consider that: $$I(a,b)=\int_{0}^{2\pi}\sqrt{a^2\sin^2\theta+b^2\cos^2\theta}\,d\theta = 4|b|\cdot E\left(\sqrt{1-\frac{a^2}{b^2}}\right)\tag{1}$$ where $E$ is the complete elliptic integral of the second kind, satisfying: $$\frac{dE(k)}{dk}=\frac{E(k)-K(k)}{k},\tag{2}$$ then consider that the given integral equals: $$a\cdot\frac{\partial}{\partial b}I(a,b).\tag{3}$$