Example. Calculate the integral

$$
\int_{0}^{0.1} \cos \left(100 x^{2}\right) d x
$$

with accuracy $\alpha=0.001$.
Solution.

1. We expand the integrand function into a Taylor series in powers of $x:$

$$
\begin{aligned}
\cos \left(100 x^{2}\right)=1-\frac{\left(10^{2} x^{2}\right)^{2}}{2!}+ & \frac{\left(10^{2} x^{2}\right)^{4}}{4!}-\frac{\left(10^{2} x^{2}\right)^{6}}{6!}+\ldots= \\
& =\sum_{n=0}^{\infty}(-1)^{n} \frac{\left(10^{2} x^{2}\right)^{2 n}}{(2 n)!}=\sum_{n=0}^{\infty}(-1)^{n} \frac{10^{4 n} x^{4 n}}{(2 n)!}
\end{aligned}
$$

The expansion is valid for all $x$.

2. We integrate the obtained series term by term:

$$
\begin{aligned}
\int_{0}^{0.1} \cos \left(10^{2} x^{2}\right) d x= & \int_{0}^{0.1} \sum_{n=0}^{\infty}(-1)^{n} \frac{10^{4 n} x^{4 n}}{(2 n)!} d x=\sum_{n=0}^{\infty}(-1)^{n} \frac{10^{4 n}}{(2 n)!} \int_{0}^{0.1} x^{4 n} d x= \\
& =\left.\sum_{n=0}^{\infty}(-1)^{n} \frac{10^{4 n}}{(2 n)!} \frac{x^{4 n+1}}{4 n+1}\right|_{0} ^{0.1}=\frac{1}{10} \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(4 n+1)(2 n)!}
\end{aligned}
$$

3. We estimate the remainder of the series. Since the series is alternating,

$$
a_{n}=\frac{1}{10(4 n+1)(2 n)!}>a_{n+1}=\frac{1}{10(4 n+5)(2 n+2)!}
$$

and $\lim _{n \rightarrow \infty} a_{n}=0$, the inequality

$$
\left|R_{n}\right| \leq a_{n+1}=\frac{1}{10(4 n+5)(2 n+2)!}
$$

is valid. To compute the integral with the given accuracy, it is sufficient to take two terms of the series, since

$$
R_{1} \leq \frac{1}{10 \cdot 9 \cdot 4!}<0.001
$$

4. Performing the calculations, we get

$$
\int_{0}^{0.1} \cos \left(100 x^{2}\right) d x \cong \frac{1}{10}-\frac{1}{10 \cdot 5 \cdot 2!} \cong 0.090
$$

Answer. $\int_{0}^{0.1} \cos \left(100 x^{2}\right) d x \cong 0.090 \pm 0.001$.

Conditions of the Problems. Compute the integral with accuracy $\alpha=0.001$.

1. $\int_{0}^{1} e^{-x^{2}} d x$.
2. $\int_{0}^{1} \sin x^{2} d x$.
3. $\int_{0}^{0.5} \frac{1}{1+x^{4}} d x$.
4. $\int_{0}^{0.5} \sqrt{1+x^{3}} d x$.
5. $\int_{0}^{0.5} e^{-x^{2} / 4} d x$.
6. $\int_{0}^{2 / 3} \frac{1}{\sqrt[3]{1+x^{5}}} d x$.
7. $\int_{0}^{1} \frac{1}{\sqrt{1+x^{4}}} d x$.
8. $\int_{0}^{1} \sqrt[3]{x} \cos x d x$.
9. $\int_{0}^{\sqrt{3} / 3} x^{3} \operatorname{arctg} x d x$.
10. $\int_{0}^{1} \frac{1}{\sqrt[3]{8+x^{3}}} d x$.

Answers. 1. $S \cong 0.747$. 2. $S \cong 0.310$. 3. $S \cong 0.494$. 4. $S \cong 0.190$. 5. $S \cong 0.508$. $6 . S \cong 0.490$. $7 . S \cong 0.662$. $8 . S \cong 0.608$. $\quad 9 . S \cong 0.012$. 10. $S \cong 0.495$.

Chapter 11

## DIFFERENTIAL EQUATIONS

In the study of the topic DIFFERENTIAL EQUATIONS, you will become familiar with various types of first and second order equations and learn methods for solving them. You will study linear equations with constant coefficients, the structure of their general solution, and methods for finding it (Euler's method, the method of undetermined coefficients, Lagrange's method).

Using the package SOLVER.VM, you can compute integrals, solve systems of equations, differentiate the found solutions, and perform numerical calculations (for example, when solving the Cauchy problem), as well as check the results obtained by you.

## 11.1. Concept of a Solution

Problem Statement. Prove that the function $y=y(x)$ satisfies the differential equation $F\left(x, y, y^{\prime}\right)=0$.

PLAN OF SOLUTION.

To prove that the function $y=y(x)$ satisfies the equation $F\left(x, y, y^{\prime}\right)=0$, it is sufficient to compute the derivative $y^{\prime}(x)$, substitute $y(x)$ and $y^{\prime}(x)$ into this equation, and verify that an identity is obtained, i.e., $F\left(x, y(x), y^{\prime}(x)\right) \equiv 0$ for all admissible $x$.