Задача 61693 Интеграл (ctg(x))^6(cos(x))^4...

61850d4e95c4af15c10aa7c9

21.11.2021 19:00:31

Интеграл (ctg(x))^6(cos(x))^4

5f3ea7e3faf909182968ddd9

21.11.2021 21:10:37

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[m]∫ ctg^8x\cdot cosec^4xdx=∫ ctg^8x\cdot \frac{1}{sin^4x}dx=∫ ctg^8x\cdot \frac{1}{sin^2x}\cdot \frac{1}{sin^2x}dx=[/m]

[m]d(ctgx)=(ctgx)`=-\frac{1}{sin^2x}dx[/m] ⇒[m] \frac{1}{sin^2x}dx=-d(ctgx)[/m]


[m]∫ ctg^8x\cdot (1+ctg^2x)\cdot (-d(ctgx))=- ∫ctg^8xd(ctgx)- ∫ ctg^{10}xd(ctgx)=-\frac{ctg^9x}{9}-\frac{ctg^{11}x}{11}+C [/m]