57. $\int \frac{2 x d x}{1+x^{2}}$.

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57. $\int \frac{2 x d x}{1+x^{2}}$.
Solution. The integrand $\frac{2 x}{1+x^{2}}$ is a fraction, the numerator of which is the differential of the denominator: $d\left(1+x^{2}\right)=$ $=2 x d x$. Then we get

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

Here, formula II is used again. The absolute value sign is not written, as the inequality $1+x^{2}>0$ holds for any value of $x$