Задача 69540 ...

6355a2d83db1a168cec87f02

23.02.2023 16:07:13

Найти неопределенный интеграл ∫ sin^4 2x dx

5f3ea7e3faf909182968ddd9

25.02.2023 10:44:53

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[m]sin^22x=\frac{1-cos4x}{2}[/m]

[m]sin^4x=(\frac{1-cos4x}{2})^2=\frac{1-2cos4x+cos^24x}{4}=\frac{1-2cos4x+\frac{1+cos8x}{2}}{4}=\frac{3}{8}-\frac{1}{2}cos4x+\frac{1}{8}cos8x[/m]


[m] ∫ sin^22xdx= ∫ (\frac{3}{8}-\frac{1}{2}cos4x+\frac{1}{8}cos8x)dx=\frac{3}{8}x-\frac{1}{8}sin4x+\frac{1}{64}(-sin8x)+C[/m]