Example 1.2. Find $\int \frac{x d x}{1+x^{2}}$.
Solution. This integral could be found using the same technique as in Example 1.1. However, we will show how the integrand is formally transformed after introducing a new variable of integration. Introduce a new variable of integration $t$, related to the old variable by the equation $t=1+x^{2}$. Next, we transform the integrand to the form of the right-hand side of (1.9) in accordance with the remarks noted after Problem 1.1. We have

$$
d t=d\left(1+x^{2}\right)=\left(1+x^{2}\right)^{\prime} d x=2 x d x \Rightarrow x d x=\frac{1}{2} d t
$$

Thus, in accordance with (1.9)

$$
\int \frac{x d x}{1+x^{2}}=\frac{1}{2} \int \frac{d t}{t}=\frac{1}{2} \ln |t|+C=\frac{1}{2} \ln \left(1+x^{2}\right)+C
$$