Задача 55600 ...

5f3ea7e3faf909182968ddd9

15.11.2020 17:40:58

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[m]sin\frac{x}{2}\cdot cos\frac{x}{4}=\frac{sin(\frac{x}{2}+\frac{x}{4})+sin(\frac{x}{2}-\frac{x}{4})}{2}=\frac{sin\frac{3x}{4}+sin\frac{x}{4}}{2}=[/m]

[m] ∫ sin\frac{x}{2}\cdot cos\frac{x}{4} dx= ∫\frac{sin\frac{3x}{4}+sin\frac{x}{4}}{2}dx=[/m]

[m]=\frac{1}{2} ∫ sin\frac{3x}{4}dx+\frac{1}{2} ∫ sin\frac{x}{4}dx=[/m]

d((3/4)x)=(3/4)dx и d(x/4)=(1/4)dx ⇒

dx=(4/3)d((3/4)x) и dx=4*d((1/4)x)

[m]=\frac{1}{2} \cdot\frac{4}{3}∫ sin\frac{3x}{4}d\frac{3x}{4}+\frac{1}{2} \cdot 4∫ sin\frac{x}{4}d\frac{x}{4}=[/m]

[m]=\frac{2}{3}\cdot (-cos\frac{3x}{4})+2 \cdot (-cos\frac{x}{4})+C[/m]