Recall that when $S_n$ denotes the $n-$interval approximation for $I={\displaystyle \underset{a}{\overset{b}{}}}f(x)dx$ from Simpson's Rule, one has

$|I-S_n|\frac{L}{180}\frac{(b-a{)}^5}{n^4}$

whenever $L|f^{(4)}(x)|$ on $(a,b).$

Determine the number of intervals $n$ for which $S_n$ approximates the exact value

$I={\displaystyle \underset{\frac{1}{10}}{\overset{\frac{1}{5}}{}}}(x^{3}logx)dx$

with an absolute error of at most $\frac{1}{3{10}^9}.$

Remark: This problem can be solved through algebraic manipulations, without the use of a calculator. So$,$ it's the kind of thing that could be asked on an exam. Give the smallest integer $n$ that is guaranteed to work, according to the error formula.

$n=$