Question

`\rm \int_(0)^(1) \sqrt{ \frac{2 - {x}^(2) }{1 - {x}^(2) } } dx \\`​

Answer 1

Substitute
`x=\sin(t)` and
`dx=\cos(t)\,dt`, and recall the definition of the elliptic integral of the second kind,

`\displaystyle E(k) = \int_0^(\pi/2) √(1 - k^2 \sin^2(\theta)) \, d\theta`

Then the integral has a value of

`\displaystyle \int_0^1 \sqrt{(2-x^2)/(1-x^2)} \, dx = \int_0^(\pi/2) √(2-\sin^2(t)) \, dt \\\\ ~~~~~~~~~~~~~~~~~~~~~ = \sqrt2 \int_0^(\pi/2) √(1 - \frac12 \sin^2(t)) \, dt \\\\ ~~~~~~~~~~~~~~~~~~~~~= \boxed{\sqrt2 E\left(\frac1{\sqrt2}\right)}`