Example 4. Find $\int \frac{d x}{\operatorname{ch}^{6} x}$.
Solution. We will use formula 1) from point $2^{\circ}$ and its consequence $1-\operatorname{th}^{2} x=\frac{1}{\operatorname{ch}^{2} x}$, obtained by dividing 1) by $\operatorname{ch}^{2} x$. We have:

$$
\begin{aligned}
& \int \frac{d x}{\operatorname{ch}^{6} x}=\int \frac{\left(\operatorname{ch}^{2} x-\operatorname{sh}^{2} x\right)^{2}}{\operatorname{ch}^{6} x} d x=\int \frac{d x}{\operatorname{ch}^{2} x}-2 \int \frac{\operatorname{sh}^{2} x}{\operatorname{ch}^{4} x} d x+\int \frac{\operatorname{sh}^{4} x}{\operatorname{ch}^{6} x} d x= \\
& =\operatorname{th} x-2 \int \operatorname{th}^{2} x d(\operatorname{th} x)+\int \operatorname{th}^{4} x d(\operatorname{th} x)=\operatorname{th} x-\frac{2}{3} \operatorname{th}^{3} x+\frac{1}{5} \operatorname{th}^{5} x+C
\end{aligned}
$$

## Exercises

Find the integrals of hyperbolic functions.

1. $\int \operatorname{sh}^{2} \frac{x}{15} \cdot \operatorname{ch}^{3} \frac{x}{15} d x$.
2. $\int\left(\operatorname{ch}^{2} 7 x+\operatorname{sh}^{2} 7 x\right) d x$.
3. $\int x^{2} \operatorname{ch} x d x$.
4. $\int \operatorname{th}^{4} x d x$.
5. $\int \frac{\operatorname{ch} x}{\sqrt{\operatorname{ch} x-1}} d x$.
6. $\int \frac{e^{x} d x}{\operatorname{ch} x+\operatorname{sh} x}$.
7. $\int \frac{e^{2 x}}{\operatorname{sh}^{4} x} d x$
8. $\int \frac{d x}{2 \operatorname{sh} x+3 \operatorname{ch} x}$.
9. $\int \operatorname{cth}^{5} x d x$.
10. $\int \frac{\operatorname{sh} x}{\sqrt{\operatorname{ch} 2 x}} d x$.

## Answers

1. $5 \operatorname{sh}^{3} \frac{x}{15}+3 \operatorname{ch}^{5} \frac{x}{15}+$ C. 2. $\frac{1}{14} \operatorname{sh} 14 x+$ C. 3. $x^{2} \operatorname{sh} x-2 x \operatorname{ch} x+$ $+2 \operatorname{sh} x+C .4$. $x-\operatorname{th} x-\frac{1}{3} \operatorname{th}^{3} x+C .5 .2 \operatorname{arctg} \sqrt{\operatorname{ch} x-1}+C$.
2. $x+C .7 .-\frac{e^{3 x}}{3 \operatorname{sh}^{3} x}+C .8 . \frac{2}{\sqrt{5}} \operatorname{arctg}\left(\frac{3 \operatorname{th} \frac{x}{2}+2}{\sqrt{5}}\right)+C$ (or $\left.\frac{2}{\sqrt{5}} \operatorname{arctg}\left(\sqrt{5} e^{x}\right)+C\right) .9 . \ln |\operatorname{sh} x|-\frac{1}{2} \operatorname{cth}^{2} x-\frac{1}{4} \operatorname{cth}^{4} x+C$.
3. $\frac{1}{\sqrt{2}} \ln (\sqrt{2} \operatorname{ch} x+\sqrt{\operatorname{ch} 2 x})+C$.

## § 7. Integration of Irrational Functions

Irrational integrands (containing radicals) are reduced to rational forms using various substitutions, depending on the type of irrational expression.

## 7.1. Quadratic Irrationalities

$1^{\circ}$. Special cases of integrals $\int \frac{d x}{\sqrt{a x^{2}+b x+c}}, \int \sqrt{a x^{2}+b x+c} d x$, $\int \frac{A x+B}{\sqrt{a x^{2}+b x+c}} d x$ require completing the square under the square root. The first integral can be reduced to table formulas 11 and 13, the second can be integrated by parts, and the third can be reduced to the first after separating the differential of the radicand in the numerator.

$2^{\circ}$ A more general integral of the form $\int \frac{P_{n}(x)}{\sqrt{a x^{2}+b x+c}} d x$, where $P_{n}(x)$ is a polynomial of degree $n$, can be integrated using the equality $\int \frac{P_{n}(x)}{\sqrt{a x^{2}+b x+c}} d x=Q_{n-1}(x) \sqrt{a x^{2}+b x+c}+\lambda \int \frac{d x}{\sqrt{a x^{2}+b x+c}}$, where $Q_{n-1}(x)$ is a polynomial of degree $(n-1)$ with unknown coefficients, and $\lambda$ is also an unknown number.

All unknown coefficients are determined from the identity obtained after differentiating the last equality.
$3^{\circ}$ Special integrals $\int R\left(x, \sqrt{a^{2}-x^{2}}\right) d x, \int R\left(x, \sqrt{x^{2}-a^{2}}\right) d x$ and $\int R\left(x, \sqrt{a^{2}+x^{2}}\right) d x$ can be reduced to trigonometric integrals using trigonometric substitutions $x=a \sin t$ or $x=a \cos t$ in the first integral, $x=\frac{a}{\sin t}$ or $x=\frac{a}{\cos t}$ in the second, and $x=a \operatorname{tg} t$ or $x=a \operatorname{ctg} t$ in the third.

Note that these substitutions are not always the best.

## Examples with Solutions