Quote:

$\displaystyle x = 2\sin\theta$

$\displaystyle dx = 3\cos\theta$

$\displaystyle 4 - (2\sin\theta)^{2} = 4 - 4 \sin^{2}\theta = 4(1 - \sin^{2}\theta) = 4 cos^{2}\theta$

$\displaystyle \int_{0}^{1}\dfrac{\sqrt{2} (2\cos \theta)}{\sqrt {4 cos^{2}\theta}}$ - What next it seems like all the thetas cancel out, so how can we evaluate it?

Even though you are out by a significant factor from the above mistake, you come to the right conclusion that the "functional" part cancels out. So how would you evaluate $\displaystyle \begin{align*} \int{ \mathrm{d}x } \end{align*}$? Hint: $\displaystyle \begin{align*} \int{ \mathrm{d}x} = \int{ 1 \, \mathrm{d}x } \end{align*}$...