741. Find the estimate $I_{1}^{*}$ of the definite integral $I=\int_{0}^{\frac{\pi}{2}} \cos x \mathrm{~d} x$ based on the data from ten trials. Take the random numbers $r_{i}$ from the first row of table 63.
Solution. We will generate 10 possible values of $X$ using the formula

$$
x_{i}=0+\frac{\pi}{2} r_{i}=1.571 r_{i}
$$

The results of the trials are presented in Table 64.

Table 64

| Trial Number $i$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $r_{i}$ | 0.100 | 0.973 | 0.253 | 0.376 | 0.520 | 0.135 | 0.863 | 0.467 | 0.354 | 0.876 |
| $x_{i}=(\pi / 2) r_{i}$ | 0.157 | 1.529 | 0.398 | 0.591 | 0.817 | 0.212 | 1.356 | 0.734 | 0.556 | 1.376 |
| $\varphi\left(x_{i}\right)=\cos x_{i}$ | 0.988 | 0.042 | 0.922 | 0.830 | 0.684 | 0.978 | 0.213 | 0.742 | 0.849 | 0.194 |

Considering that $\Sigma \varphi\left(x_{i}\right)=6.442$, we will find the desired estimate of the integral:

$$
I_{1}^{*}=(\pi / 2)(6.442 / 10)=1.01
$$