Example 2.5. $I=\int_{-1}^{1} x|x| d x$.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. 

However, since the text provided is already in a form that is commonly used in English for mathematical expressions, the translation is essentially the same:

Example 2.5. $I=\int_{-1}^{1} x|x| d x$.
Solution. Since the integrand is an odd function and the limits of integration are symmetric about zero, then

$$
\int_{-1}^{1} x|x| d x=0
$$

In the following two examples, it will be demonstrated that the formal application of the Newton-Leibniz formula (without considering the integrand functions) can lead to an incorrect result.