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zad3

Posted by M in Informacje, ... | 11.02.2008 - 23:04

$a_n=\frac{1}{\Pi} \int_{-\Pi}^\Pi h(x) cos(nx)dx$

$a_n=\frac{1}{\Pi}(\int_{-\Pi}^00cos(nx) dx + \int_0^\Pi x cos(nx) dx)$

$\int sin(cx)dx = -\frac{1}{c} cos (cx)$

$\int cos(cx)dx = \frac{1}{c} sin(cx)$

[wikipedia]

$\begin{tabular}{ll}u=x & dv= cos(nx)dx \\du=1dx & v=\frac{1}{n}sin(nx)\end{tabular}$

$a_n=\frac{1}{\Pi}(0|_{-\Pi}^0 + (\frac{x}{n} sin(x) - \int \frac{1}{n} sin(nx)dx)|^\Pi_0)$

$a_n=\frac{1}{\Pi}(\frac{x}{n}sin(nx) + \frac{1}{n^2} cos(nx))|_0^\Pi$

$a_n\frac{1}{\Pi}(\frac{\Pi}{n}sin(\Pi n) + \frac{1}{n^2}cos(\Pi n)} -\frac{1}{n^2} cos(0))$

$a_n\frac{1}{\Pi}(0+\frac{1}{n^2}(-1)^n-\frac{1}{n^2})$

$a_{2k}=$

$a_{2k+1}=$