Example. Calculate the definite integral

$$
\int_{0}^{2 \pi} \sin ^{4} 3 x \cos ^{4} 3 x d x
$$
Solution. Applying the formulas for reducing the degree, we have $\int_{0}^{2 \pi} \sin ^{4} 3 x \cos ^{4} 3 x d x=2^{-4} \int_{0}^{2 \pi}(2 \sin 3 x \cos 3 x)^{4} d x=$

$$
\begin{aligned}
&= 2^{-4} \int_{0}^{2 \pi} \sin ^{4} 6 x d x=2^{-6} \int_{0}^{2 \pi}(1-\cos 12 x)^{2} d x= \\
&= 2^{-6} \int_{0}^{2 \pi} d x-2^{-5} \int_{0}^{2 \pi} \cos 12 x d x+2^{-6} \int_{0}^{2 \pi} \cos ^{2} 12 x d x= \\
&= 2^{-5} \pi+2^{-7} \int_{0}^{2 \pi}(1+\cos 24 x) d x= \\
&= 2^{-5} \pi+2^{-7} \int_{0}^{2 \pi} d x+2^{-7} \int_{0}^{2 \pi} \cos 24 x d x=2^{-5} \pi+2^{-6} \pi=\frac{3 \pi}{64} \\
& \text { Answer. } \quad \int_{0}^{2 \pi} \sin ^{4} 3 x \cos ^{4} 3 x d x=\frac{3 \pi}{64}
\end{aligned}
$$

Conditions of the Problems. Calculate the definite integrals.

1. $\int_{0}^{\pi} 2^{3} \sin ^{2} x \cos ^{2} x d x$.
2. $\int_{0}^{2 \pi} 2^{3} \sin ^{2} x \cos ^{4} x d x$.
3. $\int_{0}^{\pi} \sin ^{2} \frac{x}{4} \cos ^{2} \frac{x}{4} d x$.
4. $\int_{0}^{\pi / 2} 2^{5} \cos ^{6} x d x$.
5. $\int_{0}^{\pi / 2} 2^{5} \sin ^{6} x d x$.
6. $\int_{0}^{\pi / 2} 2^{8} \sin ^{4} x \cos ^{4} x d x$.
7. $\int_{0}^{\pi / 2} \sin ^{2} x \cos ^{6} x d x$.
8. $\int_{0}^{\pi / 2} \sin ^{2 n} x d x$.
9. $\int_{0}^{\pi / 2} \cos ^{2 m} x d x$.
10. $\int_{0}^{\pi / 2} \sin ^{2 n} x \cos ^{2 m} x d x$.

$\begin{array}{llllllll}\text { Answers. } & \text { 1. } \pi . & 2 . \pi . & \text { 3. } \pi . & \text { 4. } 5 \pi . & \text { 5. } 5 \pi . & \text { 6. } 3 \pi . & \text { 7. } \pi \text {. }\end{array}$
8. $\frac{(2 n-1)!!}{(2 n)!!} \frac{\pi}{2}$.
9. $\frac{(2 m-1)!!}{(2 m)!!} \frac{\pi}{2}$.
10. $\frac{(2 n-1)!!(2 m-1)!!}{(2 n+2 m)!!} \frac{\pi}{2}$.

## 8.5. Integration of Expressions $R\left(x, \sqrt[p]{\frac{a x+b}{c x+d}}, \sqrt[q]{\frac{a x+b}{c x+d}}, \ldots\right)$

Problem Statement. Calculate the definite integral

$$
\int_{\alpha}^{\beta} R\left(x, \sqrt[p]{\frac{a x+b}{c x+d}}, \sqrt[a]{\frac{a x+b}{c x+d}}, \cdots\right) d x
$$

Plan of Solution. 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 arrive at integrals of rational functions.