Find the range of `f(x) = (sin^(-1) x)^(2) + 2pi cos^(-1) x + pi^(2)`

Explanation

`f(x) = (sin^(-1) x)^(2) + 2pi ((pi)/(2) - sin^(-1) x) + pi^(2)`
`= (sin^(-1) x)^(2) + pi^(2) - 2pi sin^(-1) x + pi^(2)`
`= (sin^(-1) x - pi)^(2) + pi^(2)`
`f_("min") = ((pi)/(2) - pi)^(2) + pi^(2) = (pi^(2))/(4) + pi^(2) = (5 pi^(2))/(4)`
`f_("max") = (-(pi)/(2) -pi)^(2) + pi^(2) = (9pi^(2))/(4) + pi^(2) = (13 pi^(2))/(4)`
Therefore, range is `[(5 pi^(2))/(4), (13 pi^(2))/(4)]`