Euler's Addition Theorem

Let $g(x)\equiv (1-x^2)(1-k^2x^2)$. Then

$$\int_0^a {dx\over \sqrt{g(x)}}+\int_0^b {dx\over \sqrt{g(x)}} = \int_0^c {dx\over \sqrt{g(x)}},$$

where

$$c\equiv {b\sqrt{g(a)}+a\sqrt{g(b)}\over \sqrt{1-k^2a^2b^2}}.$$