Найти определенный интеграл Интеграл 2pi cos(x/4)^8 dx 0. Created by lyanova1. algebra-ru

4)^8 dx 0...

lyanova1 @lyanova1

July 2022 1 6 Report

Найти определенный интеграл
Интеграл 2pi cos(x/4)^8 dx
0

Answers & Comments

NNNLLL54

Verified answer

$\int cos^8\frac{x}{4}\, dx=\Big[\; cos^2\alpha =\frac{1+cos2\alpha }{2}\; \Big]=\int \Big(\frac{1+cos\frac{x}{2}}{2}\Big)^4\, dx=\\\\=\frac{1}{16}\int \Big(1+4cos^3\frac{x}{2}+6cos^2\frac{x}{2}+4cos\frac{x}{2}+(cos^2\frac{x}{2})^2\Big)dx=\\\\=\frac{1}{16}\cdot \Big(x+4\int (1-sin^2\frac{x}{2})\cdot cos\frac{x}{2}\, dx+4\int cos\frac{x}{2}\, dx+\int \Big(\frac{1+cosx}{2}\Big)^2\Big)=$

$=\frac{x}{16}+\frac{1}{4}\cdot \Big (\int cos\frac{x}{2}\, dx-2\int sin^2\frac{x}{2}\cdot d(sin\frac{x}{2})\Big)+\frac{1}{2}\cdot sin\frac{x}{2}+\\\\+\frac{1}{16\cdot 4}\int (1+2cosx+cos^2x)\, dx=\frac{x}{16}+\frac{1}{2}\cdot sin\frac{x}{2}-\frac{1}{2}\cdot \frac{sin^3\frac{x}{2}}{3}+\frac{1}{2}\cdot sin\frac{x}{2}+\\\\+\frac{1}{64}\cdot (x+2sinx+\int \frac{1+cos2x}{2}\, dx)=$

$=\frac{x}{16}+sin\frac{x}{2}-\frac{1}{6}\cdot sin^3\frac{x}{2}+\frac{x}{64}+\frac{1}{32}\cdot sinx+\frac{1}{128}\, x+\frac{1}{256}\, sin2x+C\; ;\\\\\\\int \limits _0^{2\pi }cos^8\frac{x}{4}\, dx=(\frac{x}{16}+sin\frac{x}{2}-\frac{1}{6}\cdot sin^3\frac{x}{2}+\frac{x}{64}+\frac{1}{32}\cdot sinx+\frac{1}{128}\, x+\frac{1}{256}\, sin2x)\Big|_0^{2\pi }=\\\\=\frac{\pi}{8}+\frac{\pi }{32}+\frac{\pi}{64}=\frac{11\pi}{64}$

0 votes Thanks 0

Answers & Comments

Verified answer

b8d7e336a00e6a9a0f22e79bb91514c1

In his book Jack, _famous American scientist, described_main process of___human ...

aabbccdd kakie tipy gamet poluchaetsya

napishite uravneniya himicheskih reakcij ba baoh2 bacl2 baso4

iz kolody kart 36 naugad vynimayut dve karty najti veroyatnost togo chto eto asem

chem suicid opasno dlya podrostkov i obshestva zadanie po obshestvoznaniyupo punkta

etot nebolshoj mostik za kotorym nachinalsya zapretnyj lug manil nas k sebe shem

vypisat metafory iz teksta ty mozhesh upodobit nashu chelovecheskuyu prirodu v otno

isporchennoe zrenie eto metafora

21231531720 45312525 321838322 50

рекомендуемые вопросы

Helpful Links

Helpful Social