## Problem Statement

Calculate the definite integral:

$$
\int_{-\pi}^{0} 2^{8} \sin ^{6} x \cos ^{2} x d x
$$
## Solution

$$
\begin{aligned}
& \int_{-\pi}^{0} 2^{8} \sin ^{6} x \cos ^{2} x d x=2^{8} \int_{-\pi}^{0} \frac{(1-\cos 2 x)^{3}}{2^{3}} \cdot \frac{(1+\cos 2 x)}{2} d x= \\
& =2^{4} \int_{-\pi}^{0}(1+\cos 2 x)(1-\cos 2 x)(1-\cos 2 x)^{2} d x=2^{4} \int_{-\pi}^{0}\left(1-\cos ^{2} 2 x\right)\left(1-2 \cos 2 x+\cos ^{2} 2 x\right) d x= \\
& =2^{4} \int_{-\pi}^{0}\left(1-\cos ^{2} 2 x-2 \cos 2 x+2 \cos ^{3} 2 x+\cos ^{2} 2 x-\cos ^{4} 2 x\right) d x= \\
& =2^{4} \int_{-\pi}^{0}\left(1-2 \cos 2 x+2 \cos ^{3} 2 x-\cos ^{4} 2 x\right) d x= \\
& =2^{4}\left(\int_{-\pi}^{0} d x-\int_{-\pi}^{0} \cos 2 x d(2 x)+2 \int_{-\pi}^{0} \cos ^{2} 2 x(\cos 2 x d x)-\int_{-\pi}^{0} \cos ^{4} 2 x d x\right)= \\
& =2^{4}\left(\pi-\left.\sin 2 x\right|_{-\pi} ^{0}+\int_{-\pi}^{0}\left(1-\sin ^{2} 2 x\right) d(\sin (2 x))-\frac{1}{4} \int_{-\pi}^{0}\left(1+\cos ^{0} 4 x\right)^{2} d x\right)= \\
& =2^{4}\left(\pi-0+\left.\sin 2 x\right|_{-\pi} ^{0}-\left.\frac{\sin ^{3} 2 x}{3}\right|_{-\pi} ^{0}-\frac{1}{4} \int_{-\pi}^{0}(1+\cos 4 x)^{2} d x\right)= \\
& =2^{4}\left(\pi-\frac{1}{4} \int_{-\pi}^{0}\left(1+2 \cos 4 x+\cos ^{2} 4 x\right) d x\right)=2^{4} \pi-2^{2}\left(\int_{-\pi}^{0} d x+2 \int_{-\pi}^{0} \cos 4 x d x+\frac{1}{2} \int_{-\pi}^{0}(1+\cos 8 x) d x\right)= \\
& =16 \pi-4\left(\pi+\left.\frac{1}{2} \sin 4 x\right|_{-\pi} ^{0}+\frac{1}{2} \int_{-\pi}^{0}(1+\cos 8 x) d x\right)=16 \pi-4\left(\pi+\frac{1}{2}\left(\int_{-\pi}^{0} d x+\frac{1}{8} \int_{-\pi}^{0} \cos 8 x d(8 x)\right)\right)= \\
& =16 \pi-4\left(\pi+\frac{1}{2}\left(\pi+\left.\sin 8 x\right|_{-\pi} ^{0}\right)\right)=16 \pi-4\left(\pi+\frac{1}{2}(\pi+0)\right)=16 \pi-4 \pi-2 \pi=10 \pi
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_\�\�\�\�\�\� $\% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \% \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB}$ $\%$ D1\%8B_10-12»

Categories: Kuznetsov's Problem Book Integrals Problem 10 | Integrals

Ukrainian Banner Network

- Last edited: 14:24, June 17, 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 10-13

## Material from PlusPi