Решить интегралы: 1)a=pi; b= -pi [latex] \int\limits^a_b {x*cosnx} \, dx [/latex] [latex] \int\limits^a_b {cosnx} \, dx [/latex]

[latex]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[/latex] [latex]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[/latex]