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

To solve the problem, we need to arrange the functions \( f_1(x) = \frac{1}{9 - x^2} \), \( f_2(x) = \frac{1}{9 - 2x^2} \), and \( f_3(x) = \frac{1}{9 - x^2 - x^3} \) in ascending order for \( x \in (0, 1) \) and then prove the inequality involving the integral of \( f_3(x) \).

### Step-by-Step Solution

1. **Compare the functions**:
We need to compare \( f_1(x) \), \( f_2(x) \), and \( f_3(x) \) for \( x \in (0, 1) \).

- For \( x \in (0, 1) \):
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