Example. Find the indefinite integral

$$
\int \sin ^{4} 3 x \cos ^{4} 3 x d x
$$
SOLUTION. Applying the formulas for reducing the degree, we have $\int \sin ^{4} 3 x \cos ^{4} 3 x d x=2^{-4} \int(2 \sin 3 x \cos 3 x)^{4} d x=$

$$
\begin{aligned}
& =2^{-4} \int \sin ^{4} 6 x d x=2^{-6} \int(1-\cos 12 x)^{2} d x= \\
& =2^{-6} \int d x-2^{-5} \int \cos 12 x d x+2^{-6} \int \cos ^{2} 12 x d x= \\
& =2^{-6} x-\frac{2^{-5}}{12} \sin 12 x+2^{-7} \int(1+\cos 24 x) d x= \\
& =2^{-6} x-\frac{2^{-5}}{12} \sin 12 x+2^{-7} x+2^{-7} \int \cos 24 x d x= \\
& \quad=2^{-6} x-\frac{2^{-5}}{12} \sin 12 x+2^{-7} x+\frac{2^{-7}}{24} \sin 24 x+C
\end{aligned}
$$

Answer.

$$
\int \sin ^{4} 3 x \cos ^{4} 3 x d x=\frac{3}{2^{7}} x-\frac{1}{3 \cdot 2^{7}} \sin 12 x+\frac{1}{3 \cdot 2^{10}} \sin 24 x+C
$$

## PROBLEM STATEMENT. Find the indefinite integrals.

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

Answers.

1. $\frac{x}{2}+\frac{\sin 4 x}{8}+C$.
2. $\frac{5 x}{16}+\frac{\sin 6 x}{12}+\frac{\sin 12 x}{64}-\frac{\sin ^{3} 6 x}{144}+C$.
3. $\frac{3 x}{8}+\frac{\sin x}{2}+\frac{\sin 2 x}{16}+C . \quad$ 4. $\frac{3 x}{8}-\frac{\sin 8 x}{16}+\frac{\sin 16 x}{128}+C$.
4. $\frac{5 x}{16}-\frac{\sin 4 x}{8}+\frac{3 \sin 8 x}{128}+\frac{\sin ^{3} 4 x}{96}+C . \quad 6 . \frac{x}{8}-\frac{\sin 32 x}{256}+C$.
5. $\frac{x}{16}-\frac{\sin 4 x}{64}+\frac{\sin ^{3} 2 x}{48}+C . \quad 8 . \frac{x}{16}-\frac{\sin 4 x}{64}-\frac{\sin ^{3} 2 x}{48}+C$.
6. $\frac{3 x}{256}-\frac{\sin 4 x}{256}+\frac{\sin 8 x}{2048}+\frac{\sin ^{5} 2 x}{320}+C$.
7. $\frac{3 x}{256}-\frac{\sin 4 x}{256}+\frac{\sin 8 x}{2048}-\frac{\sin ^{5} 2 x}{320}+C$.

## 7.8. Integration of Expressions

$$
R\left(x, \sqrt[p]{\frac{a x+b}{c x+d}}, \quad q \sqrt{\frac{a x+b}{c x+d}}, \ldots\right)
$$

PROBLEM STATEMENT. Find the indefinite integral

$$
\int R\left(x, \sqrt[p]{\frac{a x+b}{c x+d}}, \sqrt[q]{\frac{a x+b}{c x+d}}, \ldots\right) d x
$$

where $R$ is a rational function and $p, q, \ldots$ are natural numbers.

SOLUTION PLAN.

1. Using the substitution

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

where $n$ is the least common denominator of the fractions $1 / p, 1 / q, \ldots$, we reduce the problem to integrals of rational functions.

2. We compute the antiderivative of the rational function of $t$ and return to the variable $x$ by substituting $t=\sqrt[n]{\frac{a x+b}{c x+d}}$.