University Calculus: Early Transcendentals (3rd Edition)

$4 \sqrt 3-\dfrac{4 \pi}{3}$
Let us consider $x = \sin \theta$ and $dx=\cos \theta d \theta$ Now, the given integral becomes: $\int \dfrac{4 \sin^2 \theta \cos \theta d \theta }{(1- \sin^2 \theta)^{3/2}} = 4\int_0^{\pi/3} (\sec^2 \theta -1) d \theta$ Now, integrate and use $x = \sin \theta$: $[4(\tan \theta -\theta)+C]_0^{\pi/3}=4 \sqrt 3-\dfrac{4 \pi}{3}$