If $f\left(x\right)\=1\+\sin \left(\sqrt{x}\ln \left(x\+1\right)\right)$, write $\frac{h^4}{180}$ $\underset{x}{\max }$$\(\|f″\|\)$ $\left(b-a\right)$, the error bound in Table 6.7.1, as $\frac{{\left(b-a\right)}^5}{180 n^4}$$\underset{x}{\max }$$\(\|f″\|\)$ and solve the inequality $\frac{{\left(4-1\right)}^5}{180 n^4}M\le {10}^{-4}$ for $n$, where $\underset{x}{M\=\max }$$\left(\left|f^{\left(4\right)}\right|\right)\doteq 1.0443$. This gives $n\doteq \pm 10.9$, so the appropriate value is $n\=12$. However, for $n\=10$, Simpson's rule actually approximates $\mathrm{\λ}$ with an error no worse than ${10}^{-4}$.

Determine $M$, the maximum of the absolute value of the fourth derivative of the integrand over the interval of integration.

With $a\=1\,b\=4\,M\doteq 1.0443$, solve the inequality $\frac{{\left(b-a\right)}^5}{180 n^4}$$\underset{x}{\max }$$\(\|f^{\left(4\right)}\|\)$$\le {10}^{-4}$ for $n$.

The appropriate choice of $n$ is the first positive even integer greater than $10.9$. Hence, $n\=12$ guarantees that Simpson's rule will approximate $\mathrm{\λ}$ with an error of no more than ${10}^{-4}$. From Example 6.7.1, take $\mathrm{\λ}$ to be the number $5.078061188$. To determine the actual value of $n$ for which Simpson's rule approximates $\mathrm{\λ}$ with the desired accuracy, use the ApproximateInt command as per Table 6.7.6(a).

By experiment, it is determined that $n\=10$ is the smallest value of $n$ for which Simpson's rule approximates $\mathrm{\λ}$ with an error no worse than ${10}^{-4}$.

Note that for Simpson's rule, the ApproximateInt command defaults to the subinterval form of the partitiontype option. This form of the rule evaluates the integrand at $2 k\+1$ points, thereby obtaining a much greater accuracy. To keep the number of function evaluations to $k\+1$, it is essential to include the normal form of the partitiontype option.