746. Find the estimate $I_{2}^{*}$ of the integral $I=\int_{0}^{1} \mathrm{e}^{x} \mathrm{~d} x$.
Solution. Since $\mathbf{e}^{x}=1+x+\ldots$, we will take the function $f(x)=C(1+x)$ as the density of the "auxiliary" random variable $\boldsymbol{X}$. From the condition $C \int_{0}(1+x) d x=1$, we find $C=2 / 3$. Thus, $f(x)=(2 / 3)(x+1)$.

Let's write the required integral as follows:

$$
I=\int_{0}^{1} \frac{\mathrm{e}^{x}}{(2 / 3)(x+1)} \cdot(2 / 3)(x+1) \mathrm{d} x
$$

Thus, the integral $I$ is represented as the expected value of the function $\frac{\mathrm{e}^{x}}{(2 / 3)(x+1)}$. As the desired estimate, we will use the sample mean (for simplicity, we will limit ourselves to ten trials):

$$
l_{i}^{*}=\frac{1}{10} \sum_{i=1}^{10} \frac{\mathrm{e}^{x_{i}}}{(2 / 3)\left(x_{i}+1\right)}=0.15 \sum_{i=1}^{10} \frac{\mathrm{e}^{x_{i}}}{x_{i}+1}
$$

where $x_{i}$ are possible values of $X$, which need to be generated according to the known density $f(x)=(2 / 3)(x+1)$. According to Rule 2 (see § 3), we have

$$
(2 / 3) \int_{0}^{x_{i}}(x+1) d x=r_{i}
$$

From this, we find the explicit formula for generating possible values of $X$:

$$
x_{i}=\sqrt{1+3 r_{i}}-1
$$

Table 66 provides the results of 10 trials.

![](https://cdn.mathpix.com/cropped/2024_05_22_1d1b16277022f98e68adg-327.jpg?height=39&width=250&top_left_y=1278&top_left_x=938)

| Trial Number $i$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |

By summing the numbers in the last row of Table 66, we get $\sum_{i=1}^{10} \frac{\mathrm{e}^{x_{i}}}{x_{i}+1} = 11.42$. The desired estimate, according to (*), is $l_{2}^{*}=0.15 \cdot 11.42=1.713$. 747. Find the estimates $I_{2}^{*}$ of the definite integrals:
a) $\int_{1}^{3}(x+1) \mathrm{d} x$
b) $\int_{0}^{1} \mathrm{e}^{2 x} \mathrm{~d} x$;
c) $\int_{0}^{\pi / 2} \sin x d x$

$$
\text { d) } \int_{1}^{4} \frac{\mathrm{d} x}{(1+\sqrt{x})^{2}}
$$

Hint. Take as the "auxiliary density" $f(x)$ the functions, respectively equal to: a) $(1 / 4)x$; b) $0.5(1+2 x)$; c) $\left(8 / \pi^{1}\right) x$; d) $1.092 /(1+x)$.