Example 10. Find the integral $\int \frac{d x}{5 \cos ^{2} x+9 \sin ^{2} x}$.
Solution. The integrand does not change from replacing $\sin x$ with $(-\sin x)$, $\cos x$ with $(-\cos x)$, i.e., $R(-\sin x, -\cos x) \equiv R(\sin x, \cos x)$. We apply the substitution $\operatorname{tg} x=t$. Since in this case

$$
\sin x=\frac{t}{\sqrt{1+t^{2}}}, \cos x=\frac{1}{\sqrt{1+t^{2}}}, x=\operatorname{arctg} t, d x=\frac{d t}{1+t^{2}}
$$

then

$$
\frac{1}{5 \cos ^{2} x+9 \sin ^{2} x}=\frac{1+t^{2}}{5+9 t^{2}}, \frac{d x}{5 \cos ^{2} x+9 \sin ^{2} x}=\frac{d t}{5+9 t^{2}}
$$

Therefore,

$$
\int \frac{d x}{5 \cos ^{2} x+9 \sin ^{2} x}=\int \frac{d t}{5+9 t^{2}}=\int \frac{\frac{1}{3} d(3 t)}{(\sqrt{5})^{2}+(3 t)^{2}}=
$$

$$
=\frac{1}{3} \cdot \frac{1}{\sqrt{5}} \operatorname{arctg} \frac{3 t}{\sqrt{5}}+C=\frac{1}{3 \sqrt{5}} \operatorname{arctg} \frac{3 \operatorname{tg} x}{\sqrt{3}}+C
$$

## Problems

Find the integrals:

1. $\int \sin x \cos ^{4} x d x$.
2. $\int \sin ^{3} x \cos ^{3} x d x$.
3. $\int \sin ^{2} 5 x d x$.
4. $\int \sin ^{2} x \cos ^{2} x d x$.
5. $\int \frac{\cos ^{5} x}{\sin ^{2} x} d x$.
6. $\int \sin x \sin 3 x d x$.
7. $\int \cos 4 x \cos 2 x d x$.
8. $\int \sin 3 x \cos 2 x d x$.
9. $\int \frac{d x}{5+4 \cos x}$.
10. $\int \frac{d x}{7 \cos ^{2} x+16 \sin ^{2} x}$.

## Answers

1. $-\frac{\cos ^{5} x}{5}+$ C. 2. $-\frac{\cos ^{4} x}{4}+\frac{\cos ^{6} x}{6}+$ C. 3. $\frac{x}{2}-\frac{\sin 10 x}{20}+$ C. 4. $\frac{x}{8}-$. $-\frac{\sin 4 x}{32}+$ C. 5. $-\frac{1}{\sin x}-2 \sin x+\frac{\sin ^{3} x}{3}+$ C. 6. $\frac{\sin 2 x}{4}-\frac{\sin 4 x}{8}+$ C. 7. $\frac{1}{12} \sin 6 x+\frac{1}{4} \sin 2 x+C . \quad$ 8. $-\frac{\cos 5 x}{10}-\frac{\cos x}{2}+C . \quad$ 9. $\frac{2}{3} \operatorname{arctg}\left(\frac{1}{3} \operatorname{tg} \frac{x}{2}\right)+C$.
2. $\frac{1}{4 \sqrt{7}} \operatorname{arctg} \frac{4 \operatorname{tg} x}{\sqrt{7}}+C$.

## § 4.8. Integration of Some Irrational Functions

1. Integrals of the form

$$
\int R\left[x,\left(\frac{a x+b}{c x+d}\right)^{\frac{p_{1}}{q_{1}}},\left(\frac{a x+b}{c x+d}\right)^{\frac{p_{2}}{q_{2}}}, \ldots,\left(\frac{a x+b}{c x+d}\right)^{\frac{p_{k}}{q_{k}}}\right] d x
$$

where $R$ is a rational function and $p_{1}, q_{1}, \ldots, p_{k}, q_{k}$ are integers, are found using the substitution

$$
\frac{a x+b}{c x+d}=t^{n}
$$

where $n$ is the least common multiple of the numbers $q_{1}, q_{2}, \ldots, q_{k}$.

2. The integral of a differential binomial, i.e., the integral

$$
\int x^{m}\left(a+b x^{n}\right)^{p} d x
$$

where $m, n, p$ are rational numbers, $a$ and $b$ are non-zero constants, can be reduced to the integral of a rational function in three cases:

1) when $p$ is an integer, - by expanding into terms using the binomial formula;
2) when $\frac{m+1}{n}$ is an integer, - by the substitution $a+b x^{n}=t^{s}$, where $s$ is the denominator of the fraction $p$;
3) when $\frac{m+1}{n}+p$ is an integer, - by the substitution $a x^{-n}+b=t^{s}$.