For `x in(0,1)` arrange `f_(1)(x) = (1)/(9-x^(2)), f_(2)(x) = (1)/(9-2x^(2))` and `f_(3)(x) = (1)/(9-x^(2)-x^(3))` in ascending order and hence prove that `1/6 ln2 lt int_(0)^(1)(1)/(9-x^(2)-x^(3)) dx lt (1)/(6sqrt(2)) ln 5`.

Explanation

`:' 0 lt x^(3) lt x^(2), for all x, in (0,1) rArr x^(2)lt x^(2) + x^(3) lt 2x^(2)`
`rArr -2x^(2) lt - x^(2)-x^(3)lt-x^(2) rArr 9-2x^(2)lt9-x^(2)-x^(3)lt9-x^(2)`
`rArr (1)/(9-x^(2)) lt (1)/(9-x^(2)-x^(3)) lt (1)/(9-2x)`
`f_(1)(x) lt f_(3)(f)ltf_(2)(x)` for `x in (0,1)`
`rArr underset(0)overset(1)intf_(1)(x)dx lt underset(0)overset(1)(int)f_(3)(x)dxlt underset(0)overset(1)intf_(2)(x)dx`
`rArr underset(0)overset(1)int(dx)/(9-x^(2)) lt underset(0)overset(1)int(dx)/(9-x^(2)-x^(3))lt underset(0)overset(1)int(dx)/(9-2x^(2))`
`rArr 1/6(ln|(3+x)/(3-x)|)_(0)^(t)ltunderset(0)overset(1)int(dx)/(9-x^(2)-x^(3))lt (1)/(6sqrt(2))(ln|(3+2x)/(3-2x)|)_(0)^(1)`
`rArr 1/6ln2 lt underset(0)overset(1)(int)(1)/(9-x^(2)-x^(3)) dx lt (1)/(6sqrt(2)) ln 5`