Example 6. Find $\int \frac{d x}{\sqrt{x^{2}+3 x-4}}$.
Solution. We have $x^{2}+3 x-4=(x-1)(x+4)$. Let $\sqrt{x^{2}+3 x-4}=(x-1) t$, i.e., $(x-1)(x+4)=(x-1)^{2} t^{2}$, $x+4=(x-1) t^{2}$, $x=\frac{t^{2}+4}{t^{2}-1}$, $d x=\frac{-10 t d t}{\left(t^{2}-1\right)^{2}}$, $(x-1) t=\left(\frac{t^{2}+4}{t^{2}-1}-1\right) t=\frac{5 t}{t^{2}-1}$. In this case, $t=\frac{\sqrt{t^{2}+3 x-4}}{x-1}=\sqrt{\frac{x+4}{x-1}}$.
The given integral takes the form

$$
\int \frac{-2 d t}{t^{2}-1}=-\ln \left|\frac{t-1}{t+1}\right|+C=\ln \left|\frac{\sqrt{x+4}+\sqrt{x-1}}{\sqrt{x+4}-\sqrt{x-1}}\right|+C
$$

## 7.3. Chebyshev Substitutions

Integrals of the form $\int x^{m}\left(a+b x^{n}\right)^{p} d x$, where $m, n, p$ are rational numbers, are called integrals of a differential binomial. Such integrals can be evaluated only in the following cases:

1) if $p$ is an integer, then we should let $x=t^{k}$, where $k$ is the least common multiple of the denominators of $m$ and $n$;
2) if $\frac{m+1}{n}$ is an integer, then we should let $a+b x^{n}=t^{s}$, where $s$ is the denominator of the fraction $p$,
3) if $\frac{m+1}{n}+p$ is an integer, then we should take $a+b x^{n}=x^{n} \cdot t^{s}$, where $s$ is the denominator of the fraction $p$.

All these substitutions belong to P. L. Chebyshev and bear his name.

## Examples with Solutions