Example 1.17. Find $\int \frac{d x}{x+\sqrt{x^{2}-x+2}}$.
Solution. Since $a=1>0$, we apply the first Euler substitution. We have $\sqrt{x^{2}-x+2}=t-x$. Further,

$$
\begin{gathered}
x=\frac{t^{2}-2}{2 t-1} \cdot d x=\frac{2\left(t^{2}-t+2\right)}{(2 t-1)^{2}} d t \\
\int \frac{d x}{x+\sqrt{x^{2}-x+2}}=2 \int \frac{t^{2}-t+2}{(2 t-1)^{2} t} d t=
\end{gathered}
$$

$$
\begin{gathered}
=\int\left(\frac{4}{t}-\frac{7}{2 t-1}+\frac{7}{(2 t-1)^{2}}\right) d t= \\
=4 \ln |t|-\frac{7}{2} \ln |2 t-1|-\frac{7}{2(2 t-1)}+C
\end{gathered}
$$

Substituting $t=x+\sqrt{x^{2}-x+2}$, we finally get

$$
\begin{gathered}
\int \frac{d x}{x+\sqrt{x^{2}-x+2}}=4 \ln \left|x+\sqrt{x^{2}-x+2}\right|- \\
-\frac{7}{2} \ln \left|2 x+2 \sqrt{x^{2}-x+2}-1\right|- \\
-\frac{2}{2\left(2 x+2 \sqrt{x^{2}-x+2}-1\right)}+C
\end{gathered}
$$

Integration of binomial differentials or differential binomials

A binomial differential is an expression of the form $x^{m}\left(a+b x^{n}\right)^{p} d x$, where $m, n$, and $p$ are rational numbers, and $a$ and $b$ are arbitrary constants $(a \neq 0, b \neq 0)$. Binomial differentials can be integrated in elementary functions only in the following cases.

1) The number $p$ is an integer. Then the integral $\int x^{m}\left(a+b x^{n}\right)^{p} d x$ is rationalized by the substitution $t=\sqrt[x]{x}$, where $p$ is the least common multiple of the denominators of the rational numbers $m$ and $n$.