Example 8 (to $5^{\circ}$ ). Find $\int \frac{d x}{\sin ^{2} x+4 \cos ^{2} x}$.
Solution. It seems that the universal substitution $\operatorname{tg} \frac{x}{2}=t$ is theoretically acceptable. Let's try to save on transformations.

$$
\begin{aligned}
& \int \frac{d x}{\sin ^{2} x+4 \cos ^{2} x}=\int \frac{1}{\frac{\sin ^{2} x}{\cos ^{2} x}+4} \cdot \frac{d x}{\cos ^{2} x}=\int \frac{d(\operatorname{tg} x)}{\operatorname{tg}^{2} x+4}=\{u=\operatorname{tg} x\}= \\
& =\int \frac{d u}{u^{2}+4}=\frac{1}{2} \operatorname{arctg} \frac{u}{2}+C=\frac{1}{2} \operatorname{arctg} \frac{\operatorname{tg} x}{2}+C
\end{aligned}
$$

## Exercises

Find the integrals of trigonometric functions.

1. $\int \sin ^{3} x \cos ^{4} x d x$
2. $\int \frac{\cos ^{3} x}{\sin ^{3} x} d x$.
3. $\int \sin x \cos ^{2} 2 x d x$.
4. $\int \frac{2+3 \cos x}{3-\sin x} d x$.
5. $\int \frac{\sin 5 x}{\cos ^{5} x} d x$.
6. $\int \frac{d x}{1+\sin ^{2} x}$.
7. $\int \frac{d x}{5-3 \cos x}$.
8. $\int \frac{2-\sin x}{2+\cos x} d x$.
9. $\int \frac{\cos x d x}{(1-\cos x)^{2}}$.
10. $\int \frac{\cos x d x}{\cos ^{3} x-\sin ^{3} x}$.
11. $\int \frac{d x}{1+\sin x}$.
12. $\int \frac{1+\sin x}{1-\sin x} d x$.
13. $\int \sin ^{5} x d x$.
14. $\int \operatorname{tg}^{3} x d x$.
15. $\int \frac{\sin ^{3} x}{\cos ^{3} x} d x$.
16. $\int \frac{\cos ^{2} 3 x}{\sin 3 x} d x$.
17. $\int \frac{d x}{\cos ^{3} x \sqrt{\sin 2 x}}$.
18. $\int \operatorname{ctg}^{8} x d x$.

## Answers

1. $-\frac{1}{5} \cos ^{5} x+\frac{1}{7} \cos ^{7} x+C$. 2. $\ln |\sin x|-\frac{1}{2} \sin ^{2} x+C$.
2. $-\frac{4}{5} \cos ^{5} x+\frac{4}{3} \cos ^{4} x-\cos x+C$.
3. $-\frac{3}{2} \ln \left|\operatorname{tg}^{2} \frac{x}{2}-\frac{2}{3} \operatorname{tg} \frac{x}{2}+1\right|-\frac{1}{\sqrt{2}} \operatorname{arctg} \frac{3 \operatorname{tg} \frac{x}{2}-1}{2 \sqrt{2}}-3 \ln \left|\cos \frac{x}{2}\right|+C$.
4. $\frac{1}{4} \operatorname{tg}^{4} x-\frac{1}{2} \operatorname{tg}^{2} x-\ln |\cos x|+C$. 6. $\frac{1}{\sqrt{2}} \operatorname{arctg}(\sqrt{2} \operatorname{tg} x)+C$.
5. $\frac{1}{2} \operatorname{arctg}\left(2 \operatorname{tg} \frac{x}{2}\right)+C$. 8. $\frac{4}{\sqrt{3}} \operatorname{arctg}\left(\frac{1}{\sqrt{3}} \operatorname{tg} \frac{x}{2}\right)+C$.
6. $\frac{1}{2} \operatorname{ctg} \frac{x}{2}-\frac{1}{6} \operatorname{ctg}^{3} \frac{x}{2}+C$.
7. $\frac{\sqrt{3}}{3} \operatorname{arctg} \frac{2 \operatorname{tg} x+1}{\sqrt{3}}+\ln \frac{\sqrt[6]{\operatorname{tg}^{2} x+\operatorname{tg} x+1}}{|\sqrt[3]{\operatorname{tg} x}-1|}+C .11 . \operatorname{tg}\left(\frac{x}{2}-\frac{\pi}{4}\right)+C$.
8. $2 \operatorname{tg}\left(\frac{x}{2}+\frac{\pi}{4}\right)+C .13 . \quad-\cos x+\frac{2}{3} \cos ^{3} x-\frac{1}{5} \cos ^{5} x+C$.
9. $\ln |\cos x|+\frac{1}{2} \operatorname{tg}^{2} x+C$. 15. $\frac{1}{2} \cos ^{2} x-\ln |\cos x|+C$.
10. $\frac{1}{3}\left(\ln \left|\operatorname{tg} \frac{3 x}{2}\right|+\cos 3 x\right)+C .17 . \frac{2}{\sqrt{5}}\left(\operatorname{tg}^{2} x+5\right) \sqrt{\operatorname{tg} x}+C$.
11. $\operatorname{ctg} x-\frac{1}{3} \operatorname{ctg}^{3} x+\frac{1}{5} \operatorname{ctg}^{5} x-\frac{1}{7} \operatorname{ctg}^{7} x+x+C$.

## § 6. Integration of Hyperbolic Functions

$1^{\circ}$ Hyperbolic functions are defined by the equations

$$
\begin{aligned}
\operatorname{sh} x & =\frac{e^{x}-e^{-x}}{2}-\text { hyperbolic sine, } \\
\operatorname{ch} x & =\frac{e^{x}+e^{-x}}{2}-\text { hyperbolic cosine, } \\
\operatorname{th} x & =\frac{\operatorname{sh} x}{\operatorname{ch} x}=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}-\text { hyperbolic tangent, } \\
\operatorname{cth} x & =\frac{\operatorname{ch} x}{\operatorname{sh} x}=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}-\text { hyperbolic cotangent }
\end{aligned}
$$

Tabular integrals containing hyperbolic functions are listed in the right column of the general table (§ 1.3).

$2^{\circ}$. The integration of expressions containing hyperbolic functions can be performed analogously to trigonometric expressions due to the following formulas:

$$
\begin{array}{ll}
\text { 1) } \operatorname{ch}^{2} x-\operatorname{sh}^{2} x=1, & \text { 2) } \operatorname{sh}^{2} x=\frac{\operatorname{ch} 2 x-1}{2} \\
\text { 3) } \operatorname{ch}^{2} x=\frac{\operatorname{ch} 2 x+1}{2}, & \text { 4) } \operatorname{sh} x \cdot \operatorname{ch} x=\frac{\operatorname{sh} 2 x}{2}
\end{array}
$$

$3^{\circ}$ Integrals containing hyperbolic functions can also be considered as integrals of rational functions of the variable $u=e^{x} \cdot \int R\left(e^{x}\right) d x$

## Examples with Solutions