753. Find the estimate $I_{4}^{*}$ of the integral $I=\int_{0}^{1} \sqrt{1+x^{2}} \mathrm{~d} x$.[^1]
Solution. Since $\sqrt{1+x^{2}}=1+(1 / 2) x^{2}+\ldots(|x| < 1)$

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  | 0.100 | 0.973 | 0.253 | 0.376 | $\|0.520\|$ | $\|0.135\|$ | 0.863 | 0.467 | $\|0.354\|$ | 0.8 |
| $x_{i}^{2}$ | $0.01 Q$ | 0.947 | $\|0.064\|$ | 0.141 | $\|0.270\|$ | $\|0.018\|$ | 0.745 | 0.218 | $0.125 \mid$ | 0.767 |
| $1+x_{i}^{2}$ | 1.010 | 1.947 | $\mid 1.064$ | 1.141 | $\|1.270\|$ | $\|1.018\|$ | 1.745 | $1.218$ | $\|1.125\|$ | 1.767 |
| $\sqrt{1+x_{i}^{2}}$ | 1.005 | 1.395 | 1.032 | 1.068 | $(1.127 \mid$ | $\|1.009\|$ | 1.321 | 1.104 | $\|1.061\|$ | 1.329 |
|  |  | 1.843 | 2.000 | 1.995 | $1.984 \mid$ | $2.000 \mid$ | 1.897 | 1.990 | $\|1.997\|$ | 1.89 |

68

By summing the numbers in the last row of table 68, we find the sum 19.597, substituting which into the relation (*), we obtain the desired estimate of the integral

$$
l_{i}^{*}=(19.597 / 20)+(1 / 6)=1.145 .
$$

Note that the exact value $I=1.147$.