## Problem Statement

Calculate the area of the figure bounded by the graphs of the functions:

$$
y=\cos ^{5} x \cdot \sin 2 x, y=0,\left(0 \leq x \leq \frac{\pi}{2}\right)
$$
## Solution

Integrals $14-22$

![](https://cdn.mathpix.com/cropped/2024_05_22_aa6951b1e92815ea34b6g-42.jpg?height=691&width=1059&top_left_y=1094&top_left_x=904)

$$
\begin{aligned}
& S=\int_{0}^{\frac{\pi}{2}} \cos ^{5} x \cdot \sin 2 x d x=\int_{0}^{\frac{\pi}{2}} \cos ^{5} x \cdot 2 \sin x \cos x d x= \\
& =-2 \int_{0}^{\frac{\pi}{2}} \cos ^{6} x d(\cos x)=2 \int_{\frac{\pi}{2}}^{0} \cos ^{6} x d(\cos x)=\left.\frac{2 \cos ^{7} x}{7}\right|_{\frac{\pi}{2}} ^{0}=\frac{2}{7}-0=\frac{2}{7}
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_ $\% \mathrm{D} 0 \% 9 \mathrm{~A} \% \mathrm{D} 1 \% 83 \% \mathrm{D} 0 \% \mathrm{~B} 7 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \mathrm{\%} \% \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD}$ $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+14-22$ » Categories: Kuznetsov Problem Book Integrals Problem 14 | Integrals

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## Problem Kuznetsov Integrals 14-23

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