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August 2022 1 20 Report

Решить интегралы:

1)a=pi; b= -pi

[tex] \int\limits^a_b {x*cosnx} \, dx [/tex]

[tex] \int\limits^a_b {cosnx} \, dx [/tex]

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NNNLLL54

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$2)\quad \int _{-\pi }^{\pi }\, cos\, nx\cdot dx=\frac{1}{n}\cdot sin\, nx\, |_{-\pi }^{\pi }=\frac{1}{n}\cdot (sin\pi n-\underbrace {sin(-\pi n)}_{-sin\, \pi n})=\\\\=\frac{1}{n}\cdot (0-0)=0$

$1)\quad \int _{-\pi }^{\pi }\, x\cdot cos\, nx\cdot dx=\\\\=[\, u=x,\; du=dx,\; dv=cos\, nx\, dx,\; v=\frac{1}{n}sin\, nx\, ]=\\\\=\frac{x}{n}\cdot sin\, nx|_{-\pi }^{\pi }-\frac{1}{n}\cdot \int _{-\pi }^{\pi }\, sin\, nx\cdot dx=\\\\=0+\frac{1}{n^2}\cdot cos\, nx|_{-\pi }^{\pi }=\frac{1}{n^2}\cdot (cos\pi n-\underbrace {cos(-\pi n)}_{cos\, \pi n})=\\\\=\frac{1}{n^2}\cdot ((-1)^{n}-(-1)^{n})=0$

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pororoplol спасибо как находиться интеграл xsinnx ?

NNNLLL54 Аналогично интегралу от (x*cos nx) ? по частям.

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